Review on Factoring
Frank Ma © 2011
To factor means to rewrite an expression as a product in a
nontrivial way.
Review on Factoring
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an expression...
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an expression...
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an expression...
± ± –
+
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an ex...
± ± –
+
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an ex...
± ± –
+
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an ex...
± ± –
+
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an ex...
± ± –
+
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an ex...
± ± –
+
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an ex...
± ± –
+
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an ex...
± ± –
+
To factor means to rewrite an expression as a product in a
nontrivial way. Following are the steps to factor an ex...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor a...
Example C. Factor out the GCF.
a. 12x2y3 + 6x2y2
Review on Factoring
(the GCF is 6x2y2)
Example C. Factor out the GCF.
a. 12x2y3 + 6x2y2
Review on Factoring
(the GCF is 6x2y2)
Example C. Factor out the GCF.
a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1)
Review on Factoring
(the GCF is 6x2y2)
Example C. Factor out the GCF.
a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
Review on Facto...
b. (2x – 3)3x – 2(2x – 3)
(the GCF is 6x2y2)
Example C. Factor out the GCF.
a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3)
(the GCF is 6x2y2)
Example C. Factor...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)
(the GCF is 6x2y2...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)
(the GCF is 6x2y2...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)
(the GCF is 6x2y2...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)
(the GCF is 6x2y2...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)
(the GCF is 6x2y2...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)
(the GCF is 6x2y2...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)
(the GCF is 6x2y2...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)
(the GCF is 6x2y2...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)
(the GCF is 6x2y2...
b. (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2)
(the GCF is 6x2y2...
Factoring Trinomials and Making Lists
For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are num...
For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are num...
For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are num...
For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are num...
For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are num...
For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are num...
For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are num...
For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are num...
For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are num...
Factoring Trinomials and Making Lists
One method to determine which is which is by making lists.
Factoring Trinomials and Making Lists
One method to determine which is which is by making lists.
A list is a record of all...
Factoring Trinomials and Making Lists
One method to determine which is which is by making lists.
A list is a record of all...
Factoring Trinomials and Making Lists
One method to determine which is which is by making lists.
A list is a record of all...
Factoring Trinomials and Making Lists
One method to determine which is which is by making lists.
A list is a record of all...
Factoring Trinomials and Making Lists
One method to determine which is which is by making lists.
A list is a record of all...
Factoring Trinomials and Making Lists
One method to determine which is which is by making lists.
A list is a record of all...
Factoring Trinomials and Making Lists
One method to determine which is which is by making lists.
A list is a record of all...
Example D.
Using the given tables,
list all the u and v such that:
Factoring Trinomials and Making Lists
One method to det...
Example D.
Using the given tables,
list all the u and v such that:
Factoring Trinomials and Making Lists
One method to det...
The ac-Method
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if ...
The ac-Method
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if ...
The ac-Method
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if ...
The ac-Method
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if ...
The ac-Method
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if ...
The ac-Method
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if ...
The ac-Method
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if ...
The ac-Method
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if ...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is: ac
b
Factoring Trinomials and Making ...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is:
i.e. ac at the top, and b at the bott...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is:
i.e. ac at the top, and b at the bott...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is:
i.e. ac at the top, and b at the bott...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is:
i.e. ac at the top, and b at the bott...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is:
i.e. ac at the top, and b at the bott...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is:
i.e. ac at the top, and b at the bott...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is:
i.e. ac at the top, and b at the bott...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is:
i.e. ac at the top, and b at the bott...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is:
i.e. ac at the top, and b at the bott...
ac-Method: Given the trinomial ax2 + bx + c
with no common factor, it’s ac–table is:
i.e. ac at the top, and b at the bott...
Example H. Factor 3x2 – 4x – 20 using the ac-method.
Factoring Trinomials and Making Lists
Example H. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
Factoring Trinomial...
Example H. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–t...
Example H. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–t...
Example H. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–t...
Example H. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–t...
Example H. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–t...
Example H. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–t...
Example H. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–t...
Example H. Factor 3x2 – 4x – 20 using the ac-method.
If the trinomial is prime then we have to justify it’s prime by
showi...
Example H. Factor 3x2 – 4x – 20 using the ac-method.
If the trinomial is prime then we have to justify it’s prime by
showi...
Example I. Factor 3x2 – 6x – 20 if possible.
If it’s prime, justify that.
Factoring Trinomials and Making Lists
Example I. Factor 3x2 – 6x – 20 if possible.
If it’s prime, justify that.
a = 3, c = –20, hence ac = 3(–20) = –60,
with b ...
Example I. Factor 3x2 – 6x – 20 if possible.
If it’s prime, justify that.
a = 3, c = –20, hence ac = 3(–20) = –60,
with b ...
Example I. Factor 3x2 – 6x – 20 if possible.
If it’s prime, justify that.
a = 3, c = –20, hence ac = 3(–20) = –60,
with b ...
Example I. Factor 3x2 – 6x – 20 if possible.
If it’s prime, justify that.
a = 3, c = –20, hence ac = 3(–20) = –60,
with b ...
Example I. Factor 3x2 – 6x – 20 if possible.
If it’s prime, justify that.
a = 3, c = –20, hence ac = 3(–20) = –60,
with b ...
Example I. Factor 3x2 – 6x – 20 if possible.
If it’s prime, justify that.
a = 3, c = –20, hence ac = 3(–20) = –60,
with b ...
Example J.
a. Factor x2 + 5x + 6
Factoring Trinomials and Making Lists
Example J.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v
where uv = 6
Factoring Trinomial...
Example J.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v
where uv = 6 and u + v = 5.
Fact...
Example J.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v
where uv = 6 and u + v = 5.
Sinc...
Example J.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v
where uv = 6 and u + v = 5.
Sinc...
Example J.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v
where uv = 6 and u + v = 5.
Sinc...
Example J.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v
where uv = 6 and u + v = 5.
Sinc...
b. Factor x2 – 5x + 6
Example J.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v
where uv =...
b. Factor x2 – 5x + 6
We want (x + u)(x + v) = x2 – 5x + 6,
Example J.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 +...
b. Factor x2 – 5x + 6
We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v
where uv = 6
Example J.
a. Factor x2 + 5x +...
b. Factor x2 – 5x + 6
We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v
where uv = 6 and u + v = –5.
Example J.
a. ...
b. Factor x2 – 5x + 6
We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v
where uv = 6 and u + v = –5.
Since 6 = (1)(...
b. Factor x2 – 5x + 6
We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v
where uv = 6 and u + v = –5.
Since 6 = (1)(...
b. Factor x2 – 5x + 6
We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v
where uv = 6 and u + v = –5.
Since 6 = (1)(...
c. Factor x2 + 5x – 6
b. Factor x2 – 5x + 6
We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v
where uv = 6 and u + ...
c. Factor x2 + 5x – 6
We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6
b. Factor x2 – 5x + 6
We want (x + u)(x + v...
c. Factor x2 + 5x – 6
We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and
u + v = 5.
b. Factor x2 – 5x + 6
We wan...
c. Factor x2 + 5x – 6
We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and
u + v = 5.
Since -6 = (–1)(6) = (1)(–6)...
c. Factor x2 + 5x – 6
We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and
u + v = 5.
Since -6 = (–1)(6) = (1)(–6)...
c. Factor x2 + 5x – 6
We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and
u + v = 5.
Since -6 = (–1)(6) = (1)(–6)...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Review on Factoring
Factoring Formula
If it fits, use the Difference of Squares Formula
x2 – y2 = (x + y)(x – y).
Example ...
Ex. A. Factor the following trinomials. use any method.
If it’s prime, state so.
1. 3x2 – x – 2 2. 3x2 + x – 2 3. 3x2 – 2x...
Upcoming SlideShare
Loading in …5
×

1 1 review on factoring

1,476 views

Published on

Published in: Economy & Finance, Business
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
1,476
On SlideShare
0
From Embeds
0
Number of Embeds
39
Actions
Shares
0
Downloads
0
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

1 1 review on factoring

  1. 1. Review on Factoring Frank Ma © 2011
  2. 2. To factor means to rewrite an expression as a product in a nontrivial way. Review on Factoring
  3. 3. To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring
  4. 4. To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring I. Always pull out the greatest common factor first.
  5. 5. To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method.
  6. 6. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
  7. 7. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
  8. 8. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring A common factor of two or more quantities is a factor that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
  9. 9. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Example A. a. Since 6 = (2)(3) and 15 = (3)(5), A common factor of two or more quantities is a factor that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
  10. 10. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Example A. a. Since 6 = (2)(3) and 15 = (3)(5), so 3 is a common factor. A common factor of two or more quantities is a factor that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
  11. 11. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Example A. a. Since 6 = (2)(3) and 15 = (3)(5), so 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, 2a. A common factor of two or more quantities is a factor that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
  12. 12. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Example A. a. Since 6 = (2)(3) and 15 = (3)(5), so 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, .. A common factor of two or more quantities is a factor that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
  13. 13. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Example A. a. Since 6 = (2)(3) and 15 = (3)(5), so 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, .. d. The common factor of a(x+y), b(x+y) is (x+y). A common factor of two or more quantities is a factor that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
  14. 14. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Review on Factoring
  15. 15. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} Review on Factoring
  16. 16. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. Review on Factoring
  17. 17. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} Review on Factoring
  18. 18. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. Review on Factoring
  19. 19. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} Review on Factoring
  20. 20. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. Review on Factoring
  21. 21. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. d. GCF{x3y5, x4y6, x5y4} = Review on Factoring
  22. 22. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. d. GCF{x3y5, x4y6, x5y4} = x3y4. Review on Factoring
  23. 23. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. d. GCF{x3y5, x4y6, x5y4} = x3y4. The Extraction Law Distributive law interpreted backwards gives the Extraction Law, that is, common factors may be extracted from sums or differences. Review on Factoring
  24. 24. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. d. GCF{x3y5, x4y6, x5y4} = x3y4. The Extraction Law Distributive law interpreted backwards gives the Extraction Law, that is, common factors may be extracted from sums or differences. AB ± AC  A(B±C) Review on Factoring
  25. 25. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. d. GCF{x3y5, x4y6, x5y4} = x3y4. The Extraction Law Distributive law interpreted backwards gives the Extraction Law, that is, common factors may be extracted from sums or differences. AB ± AC  A(B±C) This procedure is also called “factoring out common factor”. To factor, the first step always is to factor out the GCF. Review on Factoring
  26. 26. Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 Review on Factoring
  27. 27. (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 Review on Factoring
  28. 28. (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) Review on Factoring
  29. 29. (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) Review on Factoring
  30. 30. b. (2x – 3)3x – 2(2x – 3) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) Review on Factoring
  31. 31. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) Review on Factoring
  32. 32. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) Review on Factoring
  33. 33. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Review on Factoring
  34. 34. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Review on Factoring
  35. 35. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Review on Factoring
  36. 36. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) Review on Factoring
  37. 37. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) Review on Factoring d. x2 – x – 6
  38. 38. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) Review on Factoring d. x2 – x – 6 Write – x as –3x + 2x = x2 – 3x + 2x – 6
  39. 39. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) Review on Factoring d. x2 – x – 6 Write – x as –3x + 2x = x2 – 3x + 2x – 6 Put them into two groups = (x2 – 3x) + (2x – 6)
  40. 40. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) Review on Factoring d. x2 – x – 6 Write – x as –3x + 2x = x2 – 3x + 2x – 6 Put them into two groups = (x2 – 3x) + (2x – 6) Take out the common factor of each = x(x – 3) + 2(x – 3)
  41. 41. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) Review on Factoring d. x2 – x – 6 Write – x as –3x + 2x = x2 – 3x + 2x – 6 Put them into two groups = (x2 – 3x) + (2x – 6) Take out the common factor of each = x(x – 3) + 2(x – 3) Take out the common (x – 3) = (x – 3)(x + 2)
  42. 42. Factoring Trinomials and Making Lists
  43. 43. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. Factoring Trinomials and Making Lists
  44. 44. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. Factoring Trinomials and Making Lists
  45. 45. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Factoring Trinomials and Making Lists
  46. 46. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists
  47. 47. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists The Basic Fact About Factoring Trinomials:
  48. 48. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists The Basic Fact About Factoring Trinomials: There are two types of trinomials, l. the ones that are factorable such as x2 + 3x + 2  (x + 2)(x + 1)
  49. 49. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists The Basic Fact About Factoring Trinomials: There are two types of trinomials, l. the ones that are factorable such as x2 + 3x + 2  (x + 2)(x + 1) ll. the ones that are prime or no factorable, such as x2 + 2x + 3  (#x + #)(#x + #) (Not possible!)
  50. 50. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists The Basic Fact About Factoring Trinomials: There are two types of trinomials, l. the ones that are factorable such as x2 + 3x + 2  (x + 2)(x + 1) ll. the ones that are prime or no factorable, such as x2 + 2x + 3  (#x + #)(#x + #) Our jobs are to determine which trinomials: 1. are factorable and factor them, (Not possible!)
  51. 51. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists The Basic Fact About Factoring Trinomials: There are two types of trinomials, l. the ones that are factorable such as x2 + 3x + 2  (x + 2)(x + 1) ll. the ones that are prime or no factorable, such as x2 + 2x + 3  (#x + #)(#x + #) Our jobs are to determine which trinomials: 1. are factorable and factor them, 2. are prime so we won’t waste time on trying to factor them. (Not possible!)
  52. 52. Factoring Trinomials and Making Lists One method to determine which is which is by making lists.
  53. 53. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”.
  54. 54. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers.
  55. 55. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 12 I II Example D. Using the given tables, list all the u and v such that: 7 9
  56. 56. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 12 I i. uv is the top number II Example D. Using the given tables, list all the u and v such that: 7 9
  57. 57. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 7 12 9 I ii. and if possible, u + v is the bottom number. i. uv is the top number II Example D. Using the given tables, list all the u and v such that:
  58. 58. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 7 We list all the possible ways to factor 12 as u*v as shown. 12 9 I 1 12 6 3 4 2 1 12 6 3 4 2 ii. and if possible, u + v is the bottom number. i. uv is the top number II Example D. Using the given tables, list all the u and v such that:
  59. 59. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 7 We list all the possible ways to factor 12 as u*v as shown. For l, the solution are 3 and 4. 12 9 I 1 12 6 3 4 2 1 12 6 3 4 2 ii. and if possible, u + v is the bottom number. i. uv is the top number II Example D. Using the given tables, list all the u and v such that:
  60. 60. Example D. Using the given tables, list all the u and v such that: Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 7 We list all the possible ways to factor 12 as u*v as shown. For l, the solution are 3 and 4. For ll, based on the list of all the possible u and v, there are no u and v where u + v = 9, so the task is impossible. 12 9 I 1 12 6 3 4 2 1 12 6 3 4 2 ii. and if possible, u + v is the bottom number. i. uv is the top number II
  61. 61. Example D. Using the given tables, list all the u and v such that: Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 7 We list all the possible ways to factor 12 as u*v as shown. For l, the solution are 3 and 4. For ll, based on the list of all the possible u and v, there are no u and v where u + v = 9, so the task is impossible. 12 9 I 1 12 6 3 4 2 1 12 6 3 4 2 ii. and if possible, u + v is the bottom number. i. uv is the top number II impossible!
  62. 62. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Factoring Trinomials and Making Lists
  63. 63. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Example G. Factor x2 – x – 6 by grouping. Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
  64. 64. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
  65. 65. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 put the four terms into two pairs = (x2 – 3x) + (2x – 6) Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
  66. 66. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 put the four terms into two pairs = (x2 – 3x) + (2x – 6) take out the GCF of each pair = x(x – 3) + 2(x – 3) Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
  67. 67. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 put the four terms into two pairs = (x2 – 3x) + (2x – 6) take out the GCF of each pair = x(x – 3) + 2(x – 3) take out the common (x – 3) = (x – 3)(x + 2) Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
  68. 68. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. II. If the table is impossible to do, then the trinomial is prime. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 put the four terms into two pairs = (x2 – 3x) + (2x – 6) take out the GCF of each pair = x(x – 3) + 2(x – 3) take out the common (x – 3) = (x – 3)(x + 2) Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
  69. 69. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. II. If the table is impossible to do, then the trinomial is prime. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 put the four terms into two pairs = (x2 – 3x) + (2x – 6) take out the GCF of each pair = x(x – 3) + 2(x – 3) take out the common (x – 3) = (x – 3)(x + 2) Here is an example of factoring a trinomial by grouping. Here is how the X–table is made from a trinomial. Factoring Trinomials and Making Lists
  70. 70. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: ac b Factoring Trinomials and Making Lists
  71. 71. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, ac b Factoring Trinomials and Making Lists
  72. 72. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, In example B, the ac-table for 1x2 – x – 6 is: ac b Factoring Trinomials and Making Lists –6 –1
  73. 73. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that uv = ac u + v = b In example B, the ac-table for 1x2 – x – 6 is: ac b Factoring Trinomials and Making Lists –6 –1 u v
  74. 74. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, ac b Factoring Trinomials and Making Lists u v
  75. 75. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, ac b Factoring Trinomials and Making Lists u v
  76. 76. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, so we write x2 – x – 6 as x2 – 3x + 2x – 6 ac b Factoring Trinomials and Making Lists u v
  77. 77. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, then factor (ax2 + ux) + (vx + c) by the grouping method. In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, so we write x2 – x – 6 as x2 – 3x + 2x – 6 ac b Factoring Trinomials and Making Lists u v
  78. 78. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, then factor (ax2 + ux) + (vx + c) by the grouping method. In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, so we write x2 – x – 6 as x2 – 3x + 2x – 6 and by grouping = (x2 – 3x) + (2x – 6) ac b Factoring Trinomials and Making Lists u v
  79. 79. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, then factor (ax2 + ux) + (vx + c) by the grouping method. In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, so we write x2 – x – 6 as x2 – 3x + 2x – 6 and by grouping = (x2 – 3x) + (2x – 6) = x(x – 3) + 2(x – 3) ac b u v Factoring Trinomials and Making Lists
  80. 80. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, then factor (ax2 + ux) + (vx + c) by the grouping method. In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, so we write x2 – x – 6 as x2 – 3x + 2x – 6 and by grouping = (x2 – 3x) + (2x – 6) = x(x – 3) + 2(x – 3) = (x – 3)(x + 2) ac b u v Factoring Trinomials and Making Lists
  81. 81. Example H. Factor 3x2 – 4x – 20 using the ac-method. Factoring Trinomials and Making Lists
  82. 82. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, Factoring Trinomials and Making Lists
  83. 83. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: –60 –4 Factoring Trinomials and Making Lists
  84. 84. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. –60 –4 Factoring Trinomials and Making Lists
  85. 85. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists
  86. 86. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20
  87. 87. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20 = (3x2 + 6x ) + (–10x – 20) put in two groups
  88. 88. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20 = (3x2 + 6x ) + (–10x – 20) put in two groups = 3x(x + 2) – 10 (x + 2) pull out common factor
  89. 89. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20 = (3x2 + 6x ) + (–10x – 20) put in two groups = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) pull out common factor Hence 3x2 – 4x – 20 = (3x – 10)(x + 2)
  90. 90. Example H. Factor 3x2 – 4x – 20 using the ac-method. If the trinomial is prime then we have to justify it’s prime by showing that no such u and v exist We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20 = (3x2 + 6x ) + (–10x – 20) put in two groups = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) pull out common factor Hence 3x2 – 4x – 20 = (3x – 10)(x + 2)
  91. 91. Example H. Factor 3x2 – 4x – 20 using the ac-method. If the trinomial is prime then we have to justify it’s prime by showing that no such u and v exist by listing all the possible u’s and v’s such that uv = ac in the table to demonstrate that none of them fits the condition u + v = b. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20 = (3x2 + 6x ) + (–10x – 20) put in two groups = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) pull out common factor Hence 3x2 – 4x – 20 = (3x – 10)(x + 2)
  92. 92. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. Factoring Trinomials and Making Lists
  93. 93. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: Factoring Trinomials and Making Lists –60 –6
  94. 94. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: We want two numbers u and v such that uv = –60 and u + v = –6. Factoring Trinomials and Making Lists –60 –6
  95. 95. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: We want two numbers u and v such that uv = –60 and u + v = –6. After failing to guess two such numbers, we check to see if it's prime by listing in order all positive u’s and v’s where uv = 60 as shown. Factoring Trinomials and Making Lists –60 –6
  96. 96. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: We want two numbers u and v such that uv = –60 and u + v = –6. After failing to guess two such numbers, we check to see if it's prime by listing in order all positive u’s and v’s where uv = 60 as shown. Factoring Trinomials and Making Lists –60 –6 601 302 203 154 125 106 Always make a list in an orderly manner to ensure the accuracy of the list.
  97. 97. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: We want two numbers u and v such that uv = –60 and u + v = –6. After failing to guess two such numbers, we check to see if it's prime by listing in order all positive u’s and v’s where uv = 60 as shown. By the table, we see that there are no u and v such that ±u and ±v combine to be –6. Hence 3x2 – 6x – 20 is prime. Factoring Trinomials and Making Lists –60 –6 601 302 203 154 125 106 Always make a list in an orderly manner to ensure the accuracy of the list.
  98. 98. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: We want two numbers u and v such that uv = –60 and u + v = –6. After failing to guess two such numbers, we check to see if it's prime by listing in order all positive u’s and v’s where uv = 60 as shown. By the table, we see that there are no u and v such that ±u and ±v combine to be –6. Hence 3x2 – 6x – 20 is prime. Factoring Trinomials and Making Lists –60 –6 601 302 203 154 125 106 Always make a list in an orderly manner to ensure the accuracy of the list. Finally for some trinomials, such as when a = 1 or x2 + bx + c, it’s easier to guess directly because it must factor into the form (x ± u) (x ± v) if it’s factorable.
  99. 99. Example J. a. Factor x2 + 5x + 6 Factoring Trinomials and Making Lists
  100. 100. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 Factoring Trinomials and Making Lists
  101. 101. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Factoring Trinomials and Making Lists
  102. 102. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) Factoring Trinomials and Making Lists
  103. 103. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x Factoring Trinomials and Making Lists
  104. 104. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, Factoring Trinomials and Making Lists
  105. 105. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  106. 106. b. Factor x2 – 5x + 6 Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  107. 107. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  108. 108. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  109. 109. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  110. 110. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  111. 111. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  112. 112. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  113. 113. c. Factor x2 + 5x – 6 b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  114. 114. c. Factor x2 + 5x – 6 We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  115. 115. c. Factor x2 + 5x – 6 We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and u + v = 5. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  116. 116. c. Factor x2 + 5x – 6 We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and u + v = 5. Since -6 = (–1)(6) = (1)(–6) = (–2)(3) =(2)(–3) b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  117. 117. c. Factor x2 + 5x – 6 We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and u + v = 5. Since -6 = (–1)(6) = (1)(–6) = (–2)(3) =(2)(–3) and –1 + 6 = 5, b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  118. 118. c. Factor x2 + 5x – 6 We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and u + v = 5. Since -6 = (–1)(6) = (1)(–6) = (–2)(3) =(2)(–3) and –1 + 6 = 5, so x2 + 5x – 6 = (x – 1)(x + 6). b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
  119. 119. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y).
  120. 120. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2
  121. 121. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2
  122. 122. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y)
  123. 123. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x
  124. 124. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1)
  125. 125. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1)
  126. 126. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1)
  127. 127. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1) c. (89)(91) =
  128. 128. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1) c. (89)(91) = (90 – 1)(90 + 1)
  129. 129. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1) c. (89)(91) = (90 – 1)(90 + 1) = 902 – 12
  130. 130. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1) c. (89)(91) = (90 – 1)(90 + 1) = 902 – 12 = 8,100 – 1 = 7,099
  131. 131. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1) c. (89)(91) = (90 – 1)(90 + 1) = 902 – 12 = 8,100 – 1 = 7,099 The factors (x + y) and (x – y) are called the conjugate of each other.
  132. 132. Ex. A. Factor the following trinomials. use any method. If it’s prime, state so. 1. 3x2 – x – 2 2. 3x2 + x – 2 3. 3x2 – 2x – 1 4. 3x2 + 2x – 1 5. 2x2 – 3x + 1 6. 2x2 + 3x – 1 8. 2x2 – 3x – 27. 2x2 + 3x – 2 15. 6x2 + 5x – 6 10. 5x2 + 9x – 2 Ex. B. Factor. Factor out the GCF, the “–”, and arrange the terms in order first. 9. 5x2 – 3x – 2 12. 3x2 – 5x + 211. 3x2 + 5x + 2 14. 6x2 – 5x – 613. 3x2 – 5x + 2 16. 6x2 – x – 2 17. 6x2 – 13x + 2 18. 6x2 – 13x + 2 19. 6x2 + 7x + 2 20. 6x2 – 7x + 2 21. 6x2 – 13x + 6 22. 6x2 + 13x + 6 23. 6x2 – 5x – 4 24. 6x2 – 13x + 8 25. 6x2 – 13x – 8 26. 4x2 – 49 27. 25x2 – 4 28. 4x2 + 9 29. 25x2 + 9 30. – 6x2 – 5xy + 6y2 31. – 3x2 + 2x3– 2x 32. –6x3 – x2 + 2x 33. –15x3 – 25x2 – 10x 34. 12x3y2 –14x2y2 + 4xy2 Review on Factoring

×