This document discusses various methods for factoring polynomials, including: 1) Factoring trinomials by finding two numbers that multiply to the constant term and add to the middle coefficient. 2) The "box method" for factoring trinomials where the leading coefficient is not 1, which involves arranging terms in a 2x2 grid. 3) Factoring the difference of two squares using the formula (a + b)(a - b).

2. Factoring Polynomials
1032
−− xx
This process is basically the REVERSE
of the distributive property.
=−+ )5)(2( xx
distributive property

3. =−− 1032
xx
In factoring you start with a polynomial
(2 or more terms) and you want to rewrite it
as a product (or as a single term)
Factoring Polynomials
)5)(2( −+ xx
Three terms
One term

4. What is a perfect square?

6. 652
++ xx We need to find factors of 6
….that add up to 5

7. Factoring Trinomials, continued...
652
++ xx 2 x 3 = 6
2 + 3 = 5
Use the numbers 2 and 3 to factor the trinomial…
Write the parenthesis, with
An “x” in front of each.
( )3)2( ++ xxWrite in the two numbers
we found above.
( )xx )(

8. 652
++ xx
You can check your work by multiplying back
to get the original answer
( )3)2( ++ xx
( )3)2( ++= xx
=+++= 6232
xxx
652
++= xx
So we factored the trinomial…
Factoring Trinomials, continued...

9. Factoring Trinomials
61
65
67
2
2
2
−+
−−
++
xx
xx
xx
Find factors of – 6 that add up to –5
Find factors of 6 that add up to 7
Find factors of – 6 that add up to 1
6 and 1
– 6 and 1
3 and –2

10. 61
65
67
2
2
2
−+
−−
++
xx
xx
xx
factors of 6 that add up to 7: 6 and 1
( )1)6( ++ xx
factors of – 6 that add up to – 5: – 6 and 1
factors of – 6 that add up to 1: 3 and – 2
( )1)6( +− xx
( )2)3( −+ xx
Factoring Trinomials

11. Factoring Trinomials
The hard case – “Box Method”
62 2
−+ xx
Note: The coefficient of x2
is different from 1. In this case it is 2
62 2
−+ xx
First: Multiply 2 and –6: 2 (– 6) = – 12
1
Next: Find factors of – 12 that add up to 1
– 3 and 4

12. Factoring Trinomials
The hard case – “Box Method”
62 2
−+ xx
1. Draw a 2 by 2 grid.
2. Write the first term in the upper left-hand corner
3. Write the last term in the lower right-hand corner.
2
2x
6−

13. Factoring Trinomials
The hard case – “Box Method”
62 2
−+ xx – 3 x 4 = – 12
– 3 + 4 = 1
1. Take the two numbers –3 and 4, and put them, complete
with signs and variables, in the diagonal corners, like this:
2
2x
6−
It does not matter which
way you do the diagonal
entries!
Find factors of – 12 that add up to 1
–3 x
4x

14. The hard case – “Box Method”
1. Then factor like this:
2
2x
6−
x3−
x4
Factor Top Row Factor Bottom Row
2
2
2x
6−
x3−
x4
x
From Left Column From Right Column
2
2x
6−
x3−
x42
x
x2
2
2x
6−
x3−
x4
x2
2
x
3−
x

15. The hard case – “Box Method”
2
2x
6−
x3−
x4
x2
2
x
3−
)32)(2(62 2
−+=−+ xxxx
Note: The signs for the bottom row
entry and the right column entry
come from the closest term that you
are factoring from.
DO NOT FORGET THE SIGNS!!
++
Now that we have factored our box we can read off
our answer:

16. The hard case – “Box Method”
2
4x
12
x16−
x3−
x
3
x4
4
=+− 12194 2
xx
Finally, you can check your work by multiplying
back to get the original answer.
Look for factors of 48 that add up to –19 – 16 and – 3
)4)(34(12194 2
−−=+− xxxx

17. Use “Box” method to factor
the following trinomials.
1. 2x2
+ 7x + 3
2. 4x2
– 8x – 21
3. 2x2
– x – 6

18. Factoring the Difference of Two
Squares
The difference of two bases being squared,
factors as the product of the sum and difference
of the bases that are being squared.
a2
– b2
= (a + b)(a – b)FORMULA:
(a + b)(a – b) = a2
– ab + ab – b2
= a2
– b2

19. Factoring the difference of two squares
Factor x2
– 4y2
Factor 16r2
– 25
(x)2
(2y)
2
(x – 2y)(x + 2y)
Now you can check the results…
(4r)
2
(5)
2
Difference
of two squares
Difference
Of two squares
(4r – 5)(4r + 5)
a2
– b2
= (a + b)(a – b)

20. The information was taken from the
following people on slideshare:
Estela, Sep 22 2013. Factorising
quadratic expressions 1http://www.slideshare.net/estelav/factorising-quadratic-expressions-1
Julia Li,http://www.slideshare.net/jagheterjuliali/ch-06-10762231
Majapamaya, Nov 13, 2013. 05 perfect
square, difference of two squareshttp://www.slideshare.net/majapamaya/05-perfect-square-difference-of-two-squares
Swart J.E, Oct 28 2013. Factoring and
Box Methodhttp://www.slideshare.net/swartzje/factoring-and-box-method