The document discusses techniques for factoring polynomials. It explains how to factor the difference and sum of two squares, perfect square trinomials, and the sum and difference of two cubes. For each type of factorization, it provides steps to follow, such as taking the square root of terms for differences of squares or cube roots for sums and differences of cubes. Examples are worked through applying the steps to factor various polynomials.

1. 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.

3. Factoring
the
Difference
of Two
Squares
The difference of two squares a2
and b2 has factors with the same
first and last terms.
a2 – b2 = (a + b)(a – b)
Take note that this form of
factoring only works when the
first and last terms of the given
binomial are perfect squares and
the operation between them is
subtraction.

4. To factor
the
difference
of two
squares:
1. Square root the first term.
2. Square root the last term.
3. Multiply the sum and
difference of the first and
last term.

5. Example: Factor the following completely.
1. x2 – y2
Solution:
1. Square root the first term.
2. Square root the last term.
3. Multiply the sum and
difference of the first and
last term.
𝑥2 = 𝑥
𝑦2 = 𝑦
𝒙 + 𝒚 𝒙 − 𝒚
= 𝒙 + 𝒚)(𝒙 − 𝒚

6. Example: Factor the following completely.
2. 𝟒𝒘 𝟐
− 𝟐𝟓
Solution:
1. Square root the first term.
2. Square root the last term.
3. Multiply the sum and
difference of the first and
last term.
𝟒𝐰2= 𝟐𝐰
𝟐𝟓 = 𝟓
𝟐𝐰 + 𝟓 𝟐𝐰 − 𝟓
= 𝟐𝐰 + 𝟓)(𝟐𝐰 − 𝟓

7. Example: Factor the following completely.
3. −𝟑𝟔 + 𝒑 𝟒
Solution:
1. Square root the first term.
2. Square root the last term.
3. Multiply the sum and
difference of the first and
last term.
𝒑 𝟒 = 𝐩 𝟐
𝟑𝟔 = 𝟔
𝐩 𝟐
+ 𝟔 𝐩 𝟐
− 𝟔
= 𝒑 𝟐
+ 𝟔)(𝐩 𝟐
− 𝟔
𝒑 𝟒
− 𝟑𝟔

8. Example: Factor the following completely.
4.𝟏𝟔𝒅 𝟐
𝒆 𝟒
− 𝟔𝟒𝒇 𝟔
𝒈 𝟖
Solution:
1. Square root the first term.
2. Square root the last term.
3. Multiply the sum and
difference of the first and last
term.
𝟏𝟔𝒅 𝟐 𝒆 𝟒 = 𝟒𝐝𝐞 𝟐
𝟔𝟒𝒇 𝟔 𝒈 𝟖
= 𝟖𝐟 𝟑
𝐠 𝟒
𝟒𝐝𝐞 𝟐
+ 𝟖𝐟 𝟑
𝐠 𝟒 𝟒𝐝𝐞 𝟐
− 𝟖𝐟 𝟑
𝐠 𝟒
= 𝟒𝐝𝐞 𝟐
+ 𝟖𝐟 𝟑
𝐠 𝟒
)(𝟒𝐝𝐞 𝟐
− 𝟖𝐟 𝟑
𝐠 𝟒

9. Factoring
Perfect
Square
Trinomials
Polynomials of the form
𝒙 𝟐
+ 𝟐𝒙𝒚 + 𝒚 𝟐
𝒂𝒏𝒅
𝒙 𝟐
− 𝟐𝒙𝒚 + 𝒚 𝟐
are perfect square
trinomials. The first and last terms
of these polynomials are perfect
squares, whereas the middle term
is twice the product of the squares
of the first and last term. These
polynomials can be expressed as
products of two binomials:

10. Factoring
Perfect
Square
Trinomials
𝒙 𝟐
+ 𝟐𝒙𝒚 + 𝒚 𝟐
= (𝒙 + 𝒚)(𝒙 + 𝒚)
𝒙 𝟐
− 𝟐𝒙𝒚 + 𝒚 𝟐
= 𝒙 − 𝒚 𝒙 − 𝒚
The first and last terms of the
factors are square roots of the
first and last terms of the
product.

11. To factor the
perfect
square
trinomials:
1. Square root the first term.
2. Square root the last term.
3.The + or – sign of the middle
term of the product will be
carried as the operation
between the terms in the
factor.

12. Example: Factor the following completely.
𝒂. 𝒙 𝟐
− 𝟒𝒙 + 𝟒
Solution:
1. Square root the first term.
2. Square root the last term.
𝒙 𝟐 = 𝒙
𝟒 = 𝟐
𝒙 − 𝟐 𝒙 − 𝟐
= 𝒙 − 𝟐)(𝒙 − 𝟐
3.The + or – sign of the middle term
of the product will be carried as
the operation between the terms
in the factor.

13. Example: Factor the following completely.
𝒃. 𝒎 𝟐
+ 𝟐𝟎𝒎 + 𝟏𝟎𝟎
Solution:
1. Square root the first term.
2. Square root the last term.
𝒎 𝟐 = 𝒎
𝟏𝟎𝟎 = 𝟏𝟎
𝒎 + 𝟏𝟎 𝒎 + 𝟏𝟎
= 𝒎 + 𝟏𝟎)(𝒎 + 𝟏𝟎
3.The + or – sign of the middle term
of the product will be carried as
the operation between the terms
in the factor.

14. Example: Factor the following completely.
𝒄. 𝒏 𝟐
− 𝟏𝟔𝒏 + 𝟔𝟒
Solution:
1. Square root the first term.
2. Square root the last term.
𝒏 𝟐 = 𝐧
𝟔𝟒 = 𝟖
𝒏 − 𝟖 𝒏 − 𝟖
= 𝒏 − 𝟖)(𝒏 − 𝟖
3.The + or – sign of the middle term
of the product will be carried as
the operation between the terms
in the factor.

15. Example: Factor the following completely.
𝒅. 𝟒𝒑 𝟐
+ 𝟐𝟎𝒑 + 𝟐𝟓
Solution:
1. Square root the first term.
2. Square root the last term.
𝟒𝒑 𝟐
= 𝟐𝐩
𝟐𝟓 = 𝟓
𝟐𝒑 + 𝟓 𝟐𝒑 + 𝟓
= 𝟐𝒑 + 𝟓)(𝟐𝒑 + 𝟓
3.The + or – sign of the middle term
of the product will be carried as
the operation between the terms
in the factor.

16. Factoring
the Sum
and
Difference
of Two
Cubes
The sum and difference of
two cubes are written in the
form of 𝒙 𝟑
+𝒚 𝟑
𝒐𝒓 𝒙 𝟑
− 𝒚 𝟑
.
These polynomials have
factors
(𝒙 + 𝒚)(𝒙 𝟐
− 𝒙𝒚 + 𝒚 𝟐
) 𝒂𝒏𝒅
(𝒙 − 𝒚)(𝒙 𝟐
+ 𝒙𝒚 + 𝒚 𝟐
),
respectively.

17. To factor
the sum and
difference of
two cubes:
1. Cube root the first term.
2. Cube root the last term.
3. “Write What You See”
4. “Square-Multiply-Square”
5. “Same-Different-End on a
Positive”
6. Write the answer.
)𝒙3
± 𝒚3
= (𝒙 ± 𝒚)(𝒙2
∓ 𝒙𝒚 + 𝒚2

18. Example: Factor the following completely.
𝟏. 𝟐𝟕𝒖 𝟑
− 𝟏
Solution:
3
𝟐𝟕𝒖 𝟑 = 𝟑𝐮
3
𝟏 = 𝟏
𝟑𝒖
= (𝟑𝒖 − 𝟏)(𝟗𝒖 𝟐
+ 𝟑𝒖 + 𝟏)
1. Cube root the first term.
2. Cube root the last term.
3. “Write What You See”
4. “Square-Multiply-Square”
5. “Same-Different-End on a Positive”
6. Write the answer.
𝟏
𝟑𝒖𝟗𝒖 𝟐 𝟏
+− +
(𝟑𝒖− 𝟏) (𝟗𝒖 𝟐
+ 𝟑𝒖 + 𝟏)

19. Example: Factor the following completely.
𝟐. 𝒚 𝟏𝟐
+ 𝒛 𝟔
Solution:
3
𝒚 𝟏𝟐
= 𝐲 𝟒
3
𝒛 𝟔 = 𝐳 𝟐
𝒚 𝟒
= (𝒚 𝟒
+ 𝒛 𝟐
)(𝒚 𝟖
− 𝒚 𝟒
𝒛 𝟐
+ 𝒛 𝟒
)
1. Cube root the first term.
2. Cube root the last term.
3. “Write What You See”
4. “Square-Multiply-Square”
5. “Same-Different-End on a Positive”
6. Write the answer.
𝒛 𝟐
𝒚 𝟒
𝒛 𝟐𝒚 𝟖 𝒛 𝟒
−+ +
(𝒚 𝟒 +𝒛 𝟐
) (𝒚 𝟖
− 𝒚 𝟒
𝒛 𝟐 +𝒛 𝟐
)

20. Example: Factor the following completely.
𝟑. 𝟔𝟒 + 𝒗 𝟏𝟓
Solution:
3
𝟔𝟒 = 𝟒
3
𝒗 𝟏𝟓 = 𝒗 𝟓
𝟒
= (𝟒 + 𝒗 𝟓
)(𝟏𝟔 − 𝟒𝒗 𝟓
+ 𝒗 𝟏𝟎
)
1. Cube root the first term.
2. Cube root the last term.
3. “Write What You See”
4. “Square-Multiply-Square”
5. “Same-Different-End on a Positive”
6. Write the answer.
𝒗 𝟓
𝟒𝒗 𝟓𝟏𝟔 𝒗 𝟏𝟎
−+ +
(𝟒 + 𝒗 𝟓
) (𝟏𝟔 − 𝟒𝒗 𝟓 + 𝒗 𝟏𝟎
)

21. Example: Factor the following completely.
𝟒. 𝟖𝒙 𝟑
− 𝟏
Solution:
3
𝟖𝒙 𝟑 = 𝟐𝐱
3
𝟏 = 𝟏
𝟐𝒙
= (𝟐𝒙 − 𝟏)(𝟒𝒙 𝟐
+ 𝟐𝒙 + 𝟏)
1. Cube root the first term.
2. Cube root the last term.
3. “Write What You See”
4. “Square-Multiply-Square”
5. “Same-Different-End on a Positive”
6. Write the answer.
𝟏
𝟐𝒙𝟒𝒙 𝟐 𝟏
+− +
(𝟐𝒙− 𝟏) (𝟒𝒙 𝟐
+ 𝟐𝒙 + 𝟏)

22. Seatwork: Factor the following completely.
𝟏. 𝒙 𝟐
−𝟒
𝟐. 𝟒𝒙 𝟐
−𝟏𝟎𝟎
𝟑. 𝟗𝒙 𝟐
−𝟒𝟗
𝟒. 𝒙 𝟐
−𝟑𝟔
𝟓. 𝒙 𝟐
−𝟖𝟏
𝟔. 𝒙 𝟐
+𝟖𝒙 + 𝟏𝟔
𝟕. 𝒙 𝟐
−𝟔𝒙 + 𝟗
𝟖. 𝟒𝒙 𝟐
+𝟏𝟐𝒙 + 𝟗
𝟗. 𝒙 𝟐
−𝟏𝟎𝒙 + 𝟐𝟓
𝟏𝟎. 𝟗𝒙 𝟐
+𝟐𝟒𝒙𝒚 + 𝟏𝟔𝒚 𝟐
𝟏𝟏. 𝒙 𝟑
+𝟏𝟐𝟓
𝟏𝟐. 𝒙 𝟑
−𝟐𝟕
𝟏𝟑. 𝟐𝟕𝒙 𝟑
−𝟔𝟒
𝟏𝟒. 𝒙 𝟑
+𝟖𝒚 𝟑
𝟏𝟓. 𝟔𝟒𝒙 𝟑
− 𝟏

23. Seatwork
Answers

24. Seatwork: Factor the following completely.
𝟏. 𝒙 𝟐
−𝟒
𝟐. 𝟒𝒙 𝟐
−𝟏𝟎𝟎
𝟑. 𝟗𝒙 𝟐
−𝟒𝟗
𝟒. 𝒙 𝟐
−𝟑𝟔
𝟓. 𝒙 𝟐
−𝟖𝟏
= (𝒙 𝟐)(𝒙 𝟐)+ −
= (𝟐𝒙 𝟏𝟎)(𝟐𝒙 𝟏𝟎)+ −
= (𝟑𝒙 𝟕)(𝟑𝒙 𝟕)+ −
= (𝒙 𝟔)(𝒙 𝟔)+ −
= (𝒙 𝟗)(𝒙 𝟗)+ −

25. Seatwork: Factor the following completely.
𝟔. 𝒙 𝟐
+𝟖𝒙 + 𝟏𝟔
𝟕. 𝒙 𝟐
−𝟔𝒙 + 𝟗
𝟖. 𝟒𝒙 𝟐
+𝟏𝟐𝒙 + 𝟗
𝟗. 𝒙 𝟐
−𝟏𝟎𝒙 + 𝟐𝟓
𝟏𝟎. 𝟗𝒙 𝟐
+𝟐𝟒𝒙𝒚 + 𝟏𝟔𝒚 𝟐
= (𝒙 𝟒)(𝒙 𝟒)+ +
= (𝒙 𝟑)(𝒙 𝟑)− −
= (𝟐𝒙 𝟑)(𝟐𝒙 𝟑)+ +
= (𝒙 𝟓)(𝒙 𝟓)− −
= (𝟑𝒙 𝟒𝒚)(𝟑𝒙 𝟒𝒚)+ +

26. Seatwork: Factor the following completely.
𝟏𝟏. 𝒙 𝟑
+𝟏𝟐𝟓
𝟏𝟐. 𝒙 𝟑
−𝟐𝟕
𝟏𝟑. 𝟐𝟕𝒙 𝟑
−𝟔𝟒
𝟏𝟒. 𝒙 𝟑
+𝟖𝒚 𝟑
𝟏𝟓. 𝟔𝟒𝒙 𝟑
− 𝟏
= (𝒙 + 𝟓) (𝒙 𝟐
− 𝟓𝒙 +𝟐𝟓)
= (𝒙 − 𝟑) (𝒙 𝟐
+ 𝟑𝒙 + 𝟗)
= (𝟑𝒙 − 𝟒) (𝟗𝒙 𝟐
+𝟏𝟐𝒙+ 𝟏𝟔)
= (𝒙 +𝟐𝒚) (𝒙 𝟐
− 𝟐𝒙𝒚+ 𝟒𝒚 𝟐
)
= (𝟒𝒙 − 𝟏) (𝒙 𝟐
+ 𝟒𝒙 + 𝟏)

27. Answer Skill Builder
#1 page 24, 26 and 28
and Math Skill
Milestone letters a, b
c and d pages 29 – 30.