The document provides examples of factoring sums and differences of powers of polynomials with real and complex coefficients. It demonstrates factoring polynomials using the sums and differences of cubes theorem and sums and differences of odd powers theorem. Examples factor polynomials of the form x^n - y^n, x^n + y^n, t^7 - w^7, x - y^10, and more.

1. Section 9-8
Factoring Sums and Differences of Powers

2. Warm-up
Factor over the set of polynomials with the indicated
coefficients:
x −1
4
1. Real coefficients 2. Complex coefficients

3. Warm-up
Factor over the set of polynomials with the indicated
coefficients:
x −1
4
1. Real coefficients 2. Complex coefficients
x −1
4

4. Warm-up
Factor over the set of polynomials with the indicated
coefficients:
x −1
4
1. Real coefficients 2. Complex coefficients
x −1
4
(x −1)(x +1)
2 2

5. Warm-up
Factor over the set of polynomials with the indicated
coefficients:
x −1
4
1. Real coefficients 2. Complex coefficients
x −1
4
(x −1)(x +1)
2 2
(x −1)(x +1)(x +1)
2

6. Warm-up
Factor over the set of polynomials with the indicated
coefficients:
x −1
4
1. Real coefficients 2. Complex coefficients
x −1
4
x −1
4
(x −1)(x +1)
2 2
(x −1)(x +1)(x +1)
2

7. Warm-up
Factor over the set of polynomials with the indicated
coefficients:
x −1
4
1. Real coefficients 2. Complex coefficients
x −1
4
x −1
4
(x −1)(x +1)
2 2
(x −1)(x +1)
2 2
(x −1)(x +1)(x +1)
2

8. Warm-up
Factor over the set of polynomials with the indicated
coefficients:
x −1
4
1. Real coefficients 2. Complex coefficients
x −1
4
x −1
4
(x −1)(x +1)
2 2
(x −1)(x +1)
2 2
(x −1)(x +1)(x +1)
2
(x −1)(x − (−1))
2 2

9. Warm-up
Factor over the set of polynomials with the indicated
coefficients:
x −1
4
1. Real coefficients 2. Complex coefficients
x −1
4
x −1
4
(x −1)(x +1)
2 2
(x −1)(x +1)
2 2
(x −1)(x +1)(x +1)
2
(x −1)(x − (−1))
2 2
(x −1)(x +1)(x − i)(x + i)

10. Example 1
Find the four fourth roots of 16.

11. Example 1
Find the four fourth roots of 16.
x =16
4

12. Example 1
Find the four fourth roots of 16.
x =16
4
x −16 = 0
4

13. Example 1
Find the four fourth roots of 16.
x =16
4
x −16 = 0
4
(x − 4)(x + 4) = 0
2 2

14. Example 1
Find the four fourth roots of 16.
x =16
4
x −16 = 0
4
(x − 4)(x + 4) = 0
2 2
(x − 2)(x + 2)(x − 2i)(x + 2i) = 0

15. Example 1
Find the four fourth roots of 16.
x =16
4
x −16 = 0
4
(x − 4)(x + 4) = 0
2 2
(x − 2)(x + 2)(x − 2i)(x + 2i) = 0
x = ±2, ±2i

16. Example 1
Find the four fourth roots of 16.
x =16
4
x −16 = 0
4
(x − 4)(x + 4) = 0
2 2
(x − 2)(x + 2)(x − 2i)(x + 2i) = 0
x = ±2, ±2i
The roots are ±2, ±2i

17. Sums and Differences of
Cubes Theorem

18. Sums and Differences of
Cubes Theorem
x + y = (x + y )(x − xy + y )
3 3 2 2

19. Sums and Differences of
Cubes Theorem
x + y = (x + y )(x − xy + y )
3 3 2 2
x − y = (x − y )(x + xy + y )
3 3 2 2

20. Example 2
Factor over the set of polynomials with rational
coefficients.
x − 64y
3 12

21. Example 2
Factor over the set of polynomials with rational
coefficients.
x − 64y
3 12
(x − 4y )(x + 4xy +16y )
4 2 4 8

22. Sums and Differences of
Odd Powers Theorem

23. Sums and Differences of
Odd Powers Theorem
For all x and y and for all positive integers n:

24. Sums and Differences of
Odd Powers Theorem
For all x and y and for all positive integers n:
x + y = (x + y )(x
n n n−1
−x n− 2
y+x y − ... − xy
n−3 2 n− 2
+y n−1
)

25. Sums and Differences of
Odd Powers Theorem
For all x and y and for all positive integers n:
x + y = (x + y )(x
n n n−1
−x n− 2
y+x y − ... − xy
n−3 2 n− 2
+y n−1
)
x − y = (x − y )(x
n n n−1
+x n− 2
y+x y + ... + xy
n−3 2 n− 2
+y n−1
)

26. Example 3
Factor over the set of polynomials with rational
coefficients.
t −w
7 7

27. Example 3
Factor over the set of polynomials with rational
coefficients.
t −w
7 7
(t − w )(t + t w + t w + t w + t w + tw + w )
6 5 4 2 3 3 2 4 5 6

28. Example 4
Factor over the set of polynomials with rational
coefficients.
m +n
6 6

29. Example 4
Factor over the set of polynomials with rational
coefficients.
m +n
6 6
= (m ) + (n )
2 3 2 3

30. Example 4
Factor over the set of polynomials with rational
coefficients.
m +n6 6
= (m ) + (n )
2 3 2 3
= (m + n )(m − m n + y )
2 2 4 2 2 4

31. Example 5
Factor completely over the set of polynomials with rational
coefficients.
x −y
10 10

32. Example 5
Factor completely over the set of polynomials with rational
coefficients.
x −y
10 10
= (x ) − (y )
5 2 5 2

33. Example 5
Factor completely over the set of polynomials with rational
coefficients.
x −y
10 10
= (x ) − (y )
5 2 5 2
= (x − y )(x + y )
5 5 5 5

34. Example 5
Factor completely over the set of polynomials with rational
coefficients.
x −y10 10
= (x ) − (y )
5 2 5 2
= (x − y )(x + y )
5 5 5 5
= (x − y )(x + x y + x y + xy + y )(x + y )(x − x y + x y − xy + y )
4 3 2 2 3 4 4 3 2 2 3 4

35. Homework

36. Homework
p. 605 #1-19