-This is a powerpoint inspired by one of Canva displayed presentation. - This is about Math Polynomials and good for highschoolers presentation for school. - It consists of 39 pages explaining each of the Polynomial Techniques. - Good for review or inspired powerpoint.

1. FACTORING
POLYNOMIALS(TECHNIQUES)
By

2. Quadratic Trinomial
I. a=1 pages 17-22
II. a>1 pages 23-28
Factoring Polynomials with Greatest Common
Monomial Factor
pages 5-8
Difference of 2 Squares
pages 9-12
Factoring Perfect Square Trinomials
pages 13-16
CONTENTS

3. Factoring Quadratic Trinoial
CONTENTS
Sum and Difference of Two Cubes
pages 29-32
Factoring by Grouping
pages 33-36
Exercises Answer Keys
pages 37-38

4. Factoring is the process of finding
the factors. It reverses the
multiplication process
Factoring

5. Greatest Common Monomial
Factor
The polynomial of the highest
degree
that is an exact divisor of each of
two polynomials.
The polynomial of the highest
degree
that is an exact divisor of each of
two polynomials.

6. How do we factor polynomials using the Greatest Common Monomial Factor
Method?
Given example: 20x²+12x
Find the GCF with the lowest exponent of all the terms in a polynomial.
1.
GCMF: 4X
2. Factor out the GCF from each terms.
4x(5x)+4x(3)
3. Use distributive property to factor and simplify expressions
Final Answer: 4x(5x+3)
NOTE: Remember that distributive property says that a(b+c) is = (a)(b)+(a)(c)
Greatest Common Monomial Factor

7. Greatest Common Monomial Factor Examples
9x⁹+54x⁵ 5x⁷+30x⁴-10x³ x²y⁴+xy⁶z²+x⁵y²
9x⁹+54x⁵
1. Find the GCF of Each Term
factors of: 9x⁹: (9)(1)(x)(x)(x)(x)(x)(x)(x)(x)(x)
54x⁵ : (9)(6)(x)(x)(x)(x)(x)
The GCF is: 9x⁵
2. Use the GCF to factor each terms.
9x⁹+54x⁵ = 9x⁵(x⁴)+9x⁵ (6)
3. Make use of the Distributive Property:
The final answer is 9x⁵ (x⁴+6)
5x⁷+30x⁴-10x³
1. Find the GCF of Each Term
factors of: 5x⁷: (1)(5)(x)(x)(x)(x)(x)(x)(x)
30x⁴: (5)(6)(x)(x)(x)(x)
10x³: (5)(2)(x)(x)(x)
The GCF is: 5x³
2. Use the GCF to factor each terms.
5x⁷+30x⁴-10x³ = 5x³ (x⁴)+ 5x³(6x)-5x³ (2)
3. Make use of the Distributive Property:
The final answer is 5x³ (x⁴ +6x-2)
x² y⁴ +xy⁶z² +x⁵ y²
1. Find the GCF of Each Term
factors of: x² y⁴: (x)(x)(y)(y)
xy⁶z² : (x)(y)(y)(y)(y)(y)(y)(z)(z)
x⁵ y² :(x)(x)(x)(x)(x)(y)(y)
The GCF is: xy²
2. Use the GCF to factor each terms.
x²y⁴+xy⁶z²+x⁵y²= xy²(xy²)+xy² (y⁴z²) +xy² (x⁴)
3. Make use of the Distributive Property:
The final answer is
xy²(xy² +y⁴z² +x⁴)

8. Try These.
1. 15x²-45x⁶-60x⁸
2. 88x⁵+11y²
3. abc-ab⁵d³+ab⁹c⁴

9. Difference of Two Squares
In mathematics, the difference of two squares is a squared
(multiplied by itself) number subtracted from another squared
number. Every difference of squares may be factored
according to the identity. in elementary algebra.
-Wiki

10. How do we factor polynomials using Difference of Two Squares?
The two terms must be a polynomial.
There must be a minus sign between the two
terms
Conditions for a polynomial to be DOTS:
1.
2.
The two terms must be both squares.
There must be a minus sign between the two
terms
Conditions for a polynomial to be DOTS:
1.
2.
Finding the square root of each terms.
Write each terms’ factorization as sum and
difference of the square root.
We can factor a polynomial using DoTS by:
1.
2.
Finding the square root of each terms.
Write each terms’ factorization as sum and
difference of the square root.
We can factor a polynomial using DoTS by:
1.
2.
Difference of Two Squares

11. x²- 81 9x⁶ -25y⁴
Finding the square root of each
terms
x²-81
1.
x²: (x)²
81: (9)²
2. Write each terms’ factorization as
sum and difference of the square
root.
Final Answer: (x+9)(x-9)
Finding the square root of each
terms
9x⁶-25y⁴
1.
9x⁶: (3)² (x³)²
25y⁴: (5)² (y²)²
2. Write each terms’ factorization as
sum and difference of the square
root.
Final Answer: (3x³+5y²)(3x³-5y²)
1-25q⁴
Finding the square root of each
terms
1-25q⁴
1.
1: (1)²
25q⁴: (5q²)(²)
2. Write each terms’ factorization as
sum and difference of the square
root.
Final Answer: (1+5q²)(1-5q² )
Difference of Two Squares

12. Try These.
1. 144h²-49p²
2. a²b ²– 16
3. 100g²-25

13. Perfect Square Trinomials
In mathematics, Perfect Square Trinomial is an
expression obtained from the square of binomial
equation.
The perfect square formula takes the following:
(ax)²+2abx+b² = (ax+b)²

14. Identify the square root of the first and last terms.
Determine the operation of the middle term to be used on
How do we factor?
1.
2.
separating the factored form.
3. Write the final answer as square of binomial.
x²+2xy+y²=(x+y)²
Identify the square root of the first and last terms.
Determine the operation of the middle term to be used on
How do we factor?
1.
2.
separating the factored form.
3. Write the final answer as square of binomial.
x²+2xy+y²=(x+y)²
Perfect Square Trinomials
The first and last term must be perfect squares.
There must be no minus sign before the first and last term.
If you multiply its first and last term, the result must be its
additive inverse.
Conditions for a polynomial to be a PST:
1.
2.
3.
The first and last term must be perfect squares.
There must be no minus sign before the first and last term.
If you multiply its first and last term, the result must be its
additive inverse.
Conditions for a polynomial to be a PST:
1.
2.
3.
How do we factor polynomials using Perfect Square Trinomials?

15. Identify the square root of
the first and last terms.
1.
x²: (x)²
4: (2) ²
2. Determine the operation of the
middle term to be used on
separating the factored form.
x² +4x+4
3. Write the final answer as
square of binomial.
(x+2)²
x²+4x+4
x²+4x+4 16x² +72x+81
16x² +72x+81
Identify the square root of
the first and last terms.
1.
16x ²: (4x)²
81: (9)²
2. Determine the operation of the
middle term to be used on
separating the factored form.
16x² +72x+81
3. Write the final answer as
square of binomial.
(4x +9)²
25p²-60pq+36q²
25p²-60pq+36q²
Identify the square root of
the first and last terms.
1.
25p²: (5p)²
36q²: (6q)²
2. Determine the operation of the
middle term to be used on
separating the factored form.
25p² -60pq+36q²
3. Write the final answer as
square of binomial.
(5p-6q)²

16. Try These.
1. r²+6r+9
2. 25+40x+16x²
3. (a+b)²+10(a+b)+25

17. Quadratic Trinomial (a=1)
A quadratic trinomial is a
trinomial of the form ax² +bx+c,
where a=1.

18. List all pairs of integers whose product is equal to the last term
Choose a pair, m and n, whose sum is the middle term: m+n=b
Write the factorization as x² +bx+c=(x+m)(x+n).
If the terms of trinomial do not have a factor, then terms of
binomial factor cannot have a common factor
If constant term of a trinomial is:
Factoring x² + bx +c
1.
2.
3.
Points to remember in factoring Quadratic Trinomial.
1.
2.
a. positive, the constant term of the binomials have the same signs as
the coeffecient of x in the trinomial.
b. negative, the constant terms of the binomials have opposite signs
How do we factor polynomials using Quadratic Trinomials (a=1)

19. Quadratic Trinomial (a=1) Examples
List all pairs of integers whose product is equal to the
last term
1.
last term is 10:
Factors of 10: 5 and 2, 10 and 1
2. Choose a pair, m and n, whose sum is the middle term:
m+n=b
The factors 5 and 2 is equal to the middle term 7.
3. Write the factorization as x² +bx+c=(x+m)(x+n)
Final answer is (x+5)(x+2)
1.Example: x² +7x+10
Some Factors of 10 equal to 7
-5 and -2
5 and 2
-1 and -10
1 and 10
-7
7
-11
11

20. Quadratic Trinomial (a=1) Examples
List all pairs of integers whose product is equal to the
last term
1.
last term is 14
Factors of 14: 1,2,7,14
2. Choose a pair, m and n, whose sum is the middle term:
m+n=b
The factors -2 and -7 is equal to the middle term -9
3. Write the factorization as x² +bx+c=(x+m)(x+n)
Final answer is (x-2)(x-7)
2. Example: x²-9x+14
Some Factors of 14 equal to -9x
-1 and -14
1 and 14
2 and 7
-2 and -7
-7
7
-9
9

21. Quadratic Trinomial (a=1) Examples
List all pairs of integers whose product is equal to the
last term
1.
last term is 18
Factors of 14: 1,2,3,6, 9,18
2. Choose a pair, m and n, whose sum is the middle term:
m+n=b
The factors -2 and -9 is equal to the middle term -11
3. Write the factorization as x² +bx+c=(x+m)(x+n)
Final answer is (x-2)(x-9)
3. x²-11x+18
Some Factors of 18 equal to -11x
1,18
-1,-18
-2,-9
2,9
-3, -6
3, 6
19
-19
-11
11
-9
9

22. 1. p²+14p+40
2. b²-8b+15
3.r²+r-20
Try These.
1. p²+14p+40
2. b²-8b+15
3.r²+r-20

23. Quadratic Trinomial (a>1)
A Quadratic Trinomial (a>1) is a trinomial of the
form ax² +bx+c, where a>1

24. How do we factor polynomials using Quadratic Trinomials (a=1)
Start by finding two numbers that multiply to ac and b
Use these numbers to split up the x-term.
Use grouping to factor the quadratic expression.
In general, we can use the following steps to factor a quadratic form: ax²+
bx+c
1.
2.
3.
4.
Points to remember in factoring Quadratic Trinomial.
If the terms of trinomial do not have a factor, then terms of binomial factor
cannot have a common factor
If constant term of a trinomial is:
a. positive, the constant term of the binomials have the same signs as the
coeffecient of x in the trinomial.
b. negative, the constant terms of the binomials have opposite signs

25. Quadratic Trinomial (a>1) Examples
Factors of 56 whose sum is 15
1, 56
4, 14
7,8
57
30
18
15
Start by finding two numbers that multiply to ac and
add to b.
In general, we can use the following steps to factor a
quadratic form
2x² +15x+28
1.
(2)(28)= 56
Factors of 56 that adds to 15: 7, 8
2. Use these numbers to split up the x-term.
2x² +15x+28=2x²+(7x+8x)+28
3. Use grouping to factor the quadratic equation
2x² +7x+8x+28
=(2x² +7x)+(8x+28)
= (2x+7)(x+4) is the final answer.
1.Example: 2x+15x+28
2, 28

26. Quadratic Trinomial (a>1) Examples
Factors of -24 whose sum is -10
Start by finding two numbers that multiply to ac and
add to b.
In general, we can use the following steps to factor a
quadratic form
3x² -10x-8
1.
(3)(-8)=-24
Factors of -24 that adds to -10: -12 and 2
2. Use these numbers to split up the x-term.
3x²-10x-8 = 3x²-12+2-8
3. Use grouping to factor the quadratic equation
3x² +2x-12x-8=
x(3x+2)-4(3x+2)=
(3x+2)(x-4) is the final answer.
2. Example:3x² -10x-8
-1,24
1,-24
-2,12
-12,2
-3,8
3,-8
-4,6
4,-6
23
-23
10
-10
5
-5
2
-2

27. Quadratic Trinomial (a>1) Examples
Factors of 24 whose sum is -25
-1, -24
-3, -8
-4, -6
-25
-11
-10
Start by finding two numbers that multiply to ac and
add to b.
In general, we can use the following steps to factor a
quadratic form
6x²-25x+4
1.
(6)(4)=24
Factors of 24 that adds to -25: (1)(24)
2. Use these numbers to split up the x-term.
6x²-25x+4= 6x²-x-24x+4
3. Use grouping to factor the quadratic equation
6x²-x-24+4
=x(6x-1)-4(6x-1)
(6x-1)(x-4)
= (6x-1)(x-4) is the final answer.
3.Example:6x² -25x+4
-2, -12 -14

28. Try These.
1. 12x²-29x+15
2. 7a²+53a+28
3. 6x² +37x+6

29. Sum and Difference of two
Cubes
A binomial with each term being cubed
and it has either
addition od subtraction as the middle
sign

30. a. For the binomial factor, Find the cube root of the first and last
term, copy the sign of the given.
1.
2. a. For the trinomial factor, square the first term of the binomial
factor to be put on the first term of the trinomial factor.
b. For the middle term, multiply the cuberoots.
c. square the last term of the binomial to be put on the last term of
the trinomial.
points to remember in factoring sum and difference of two cubes:
a. The sign separating the trinomials’ middle and last term must be
always addition or positive.
b. The sign separating the trinomials’ first and middle term must be
opposite to the given sign
How do we factor polynomials using the sum and difference of two cubes?
Sum and Difference of two Cubes

31. Sum and Difference of two Cubes
a+b³
a. For the binomial factor, Find the cube root of the first
and last term, copy the sign of the given.
1.
a+b³ =(a+b)
2. a. For the trinomial factor, square the first term of the
binomial factor to be put on the first term of the trinomial
factor.
b. For the middle term, multiply the cube roots.
square the last term of the binomial to be put on the last
term of the trinomial.
1a. binomial factor: (a+b)
2a. trinomial factor’ first term: (a)² = a²
b. Middle term: (a)(b)=ab
b. last term: (b)² =b²
Combine :
(a+b)(a²-ab+b²)
Hence, the final answer is
a+b³ = (a+b)(a²-ab+b²)
a. For the binomial factor, Find the cube root of
the first and last term, copy the sign of the given.
1.
y³-8=( y-2)
2. a. For the trinomial factor, square the first term of
the binomial factor to be put on the first term of the
trinomial factor.
b. For the middle term, multiply the cube roots.
c. square the last term of the binomial to be put on the
last term of the trinomial.
1a. binomial factor: (y-2)
2a. trinomial factor’ first term: (y)²
b. Middle term: (2)(y) =2y
c. last term: (2)² =4
Combine :
(y-2)(y² +2y+4)
Hence, the final answer is
y³-8=(y-2)(y² +2y+4)
y³-8 a. For the binomial factor, Find the cube root of the
first and last term, copy the sign of the given.
1.
27x³ +64y³ =(3x+4y)
2. a. For the trinomial factor, square the first term of
the binomial factor to be put on the first term of the
trinomial factor.
b. For the middle term, multiply the cube roots.
c. square the last term of the binomial to be put on the
last term of the trinomial.
1.a. binomial factor: (3x+4y)
2. a. trinomial factor’ first term: (3x)² =9x²
b. Middle term: (3x)(4y)=12xy
c. last term: (4y)² =16y²
Combine :
(3x+4y)(9x²-12xy+16y²)
Hence, the final answer is
27x³ +64y³ = (3x+4y)(9x²-12xy+16y²)
27x³ +64y³

32. Try These.
1. 27m³-125
2. 500x³ +256
3. 375-81a³

33. Factoring by Grouping
Just like it says, factoring by grouping means that
you will group terms with common factors before
factoring.

34. Step 1: Group terms with Common Monomial Factor.
Step 2: Factor out the GCF from each separate groups.
Step 3: Factor out the common binomial factor.
How do we factor polynomials using Factoring by Grouping?
Factoring by Grouping

35. 2x² +8x+3x+12
Step 1: Group terms with Common
Monomial Factor.
(2x² +8x)+(3x+12)
Step 2: Factor out the GCF from each
separate groups.
2x(x+4)+3(x+4)
Step 3: Factor out the common
binomial factor.
(2x+3)(x+4)
3x²-6x-4x+8
Step 1: Group terms with Common
Monomial Factor.
(3x²-6x)+(-4x+8)
Step 2: Factor out the GCF from each
separate groups.
3x(x-2)-4(x-2)
Step 3: Factor out the common
binomial factor.
(x-2)(3x-4)
2x³+3x-10x²-15
Step 1: Group terms with Common
Monomial Factor.
(2x³-10x²)+(3x-15)
Step 2: Factor out the GCF from each
separate groups.
2x² (x-5)+3(x-5)
Step 3: Factor out the common
binomial factor.
(x-5)(2x² +3)
Factoring by Grouping

36. Try These.
1. 5mn + 25m + 3n³ + 15n²
2. 4b³+ b² + 8b + 2
3. 28k³ − 4k²− 35k + 5

37. Answer Keys

38. 15x² (1-3x⁴-4x⁶)
Greatest Common
Monomial Factor
1.
2. 11(8x⁵ +y)
3. ab(c-b⁴d³+b⁸c⁴
DoTS
1. (12h+7p)(12h-7p)
2.(ab+4)(ab-4)
3. 25(2g+1)(2g-1)
Perfect Square
Trinomial
1.(r+3)²
2.(5+4x)²
3. (a+b+5)²
Quadratic
Trinomial a=1
1. (p+4)(p+10)
2. (b-5)(b-3)
3. (r-4)(r-5)
Quadratic Trinomial
a>1
1. (4x-3)(3x-5)
2. (a+7)(7a+4)
3. (6x+1)(x+6)
Sum and Dif. of two
cubes
1. (3m-5)(9m² +15m+25)
2. 4(5x+4)(25x²-20x+16)
3. 3(5-3a)(25+15a+9a²)
Factor by Grouping
1.(5m+3n²)(n+5)
2.(4b+1)(b² +2)
3. (7k-1)(4k²-5)

39. -Oronce,O.A.,&Mendoza,M.O.(2019).E-Math:
WorktextinMathematics.REXBookStore.
-KhanAcademy.AccessedJanuary31,2020
https://www.khanacademy.org.
-Factoring.(2019,February22).InWikipedia.URL
-OnlineLearningMath.com.AccessedJanuary31,2020.
https://www.onlinemathlearning.com/factoring-by-
grouping.html
Resources