8.5 – factoring differences of squares. recall: recall: product of a sum & a difference

8.5 – Factoring Differences of Squares

Recall:

Recall: Product of a Sum & a Difference

Recall: Product of a Sum & a Difference

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b)

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

*Use in reverse to factor!

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

*Use in reverse to factor!*a and b MUST be perfect squares!

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

*Use in reverse to factor!*a and b MUST be perfect squares!

Ex. 1 Factor each binomial below.a. n2 – 25

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

*Use in reverse to factor!*a and b MUST be perfect squares!

Ex. 1 Factor each binomial below.a. n2 – 25 (n )(n )

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

*Use in reverse to factor!*a and b MUST be perfect squares!

Ex. 1 Factor each binomial below.a. n2 – 25 (n + )(n – )

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

*Use in reverse to factor!*a and b MUST be perfect squares!

Ex. 1 Factor each binomial below.a. n2 – 25 (n + 5)(n – 5)

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

*Use in reverse to factor!*a and b MUST be perfect squares!

Ex. 1 Factor each binomial below.a. n2 – 25 (n + 5)(n – 5)b. 36x2 – 49y2

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

*Use in reverse to factor!*a and b MUST be perfect squares!

Ex. 1 Factor each binomial below.a. n2 – 25 (n + 5)(n – 5)b. 36x2 – 49y2

( x + )( x – )

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

*Use in reverse to factor!*a and b MUST be perfect squares!

Ex. 1 Factor each binomial below.a. n2 – 25 (n + 5)(n – 5)b. 36x2 – 49y2

(6x + )(6x – )

Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2

*Use in reverse to factor!*a and b MUST be perfect squares!

Ex. 1 Factor each binomial below.a. n2 – 25 (n + 5)(n – 5)b. 36x2 – 49y2

(6x + 7y)(6x – 7y)

Ex. 2 Factor each polynomial below.a. 48a3 – 12a

Ex. 2 Factor each polynomial below.a. 48a3 – 12a 12a( )

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a( a + )( a – )

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a( a + )( a – )

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + )(2a – )

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + )(2a – )

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

Ex. 2 Factor each polynomial below.a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1) b. 2x4 – 162

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

b. 2x4 – 162

2( )

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

b. 2x4 – 162

2(x4 – 81)

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

b. 2x4 – 162

2(x4 – 81)

2(x2 + )(x2 – )

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

b. 2x4 – 162

2(x4 – 81)

2(x2 + 9)(x2 – 9)

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

b. 2x4 – 162

2(x4 – 81)

2(x2 + 9)(x2 – 9)

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

b. 2x4 – 162

2(x4 – 81)

2(x2 + 9)(x2 – 9)

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

b. 2x4 – 162

2(x4 – 81)

2(x2 + 9)(x2 – 9)

2(x2 + 9)(x + )(x – )

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

b. 2x4 – 162

2(x4 – 81)

2(x2 + 9)(x2 – 9)

2(x2 + 9)(x + )(x – )

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

b. 2x4 – 162

2(x4 – 81)

2(x2 + 9)(x2 – 9)

2(x2 + 9)(x + 3)(x – 3)

Ex. 2 Factor each polynomial below.

a. 48a3 – 12a

12a(4a2 – 1)

12a(2a + 1)(2a – 1)

b. 2x4 – 162

2(x4 – 81)

2(x2 + 9)(x2 – 9)

2(x2 + 9)(x + 3)(x – 3)

Ex. 3 Solve each equation by factoring.

a. p2 – 1 = 0

Ex. 3 Solve each equation by factoring.

a. p2 – 1 = 0

(p + )(p – ) = 0

Ex. 3 Solve each equation by factoring.

a. p2 – 1 = 0

(p + 1)(p – 1) = 0

Ex. 3 Solve each equation by factoring.

a. p2 – 1 = 0

(p + 1)(p – 1) = 0

p + 1 = 0 p – 1 = 0

Ex. 3 Solve each equation by factoring.

a. p2 – 1 = 0

(p + 1)(p – 1) = 0

p + 1 = 0 p – 1 = 0

- 1 - 1 + 1 + 1

Ex. 3 Solve each equation by factoring.

a. p2 – 1 = 0

(p + 1)(p – 1) = 0

p + 1 = 0 p – 1 = 0

- 1 - 1 + 1 + 1

p = -1 p = 1

Ex. 3 Solve each equation by factoring.

a. p2 – 1 = 0

(p + 1)(p – 1) = 0

p + 1 = 0 p – 1 = 0

- 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x

Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0

(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x - 50x -50x

Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0

(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x - 50x -50x

18x3 – 50x = 0

Ex. 3 Solve each equation by factoring.

a. p2 – 1 = 0

(p + 1)(p – 1) = 0

p + 1 = 0 p – 1 = 0

- 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x

- 50x -50x

18x3 – 50x = 0

2x(9x2 – 25) = 0

Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0

(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0

2x( x + )( x – ) = 0

Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0

(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0

2x(3x + )(3x – ) = 0

Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0

(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0

2x(3x + 5)(3x – 5) = 0

Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0

(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0

2x(3x + 5)(3x – 5) = 0 2x = 0

Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0

(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0

2x(3x + 5)(3x – 5) = 0 2x = 03x + 5 = 0

Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0

(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0

2x(3x + 5)(3x – 5) = 0 2x = 03x + 5 = 0 3x – 5 = 0

Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0

(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1

p = -1 p = 1

b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0

2x(3x + 5)(3x – 5) = 0 2x = 03x + 5 = 0 3x – 5 = 0 x = 0 x = -5/3 x = 5/3