Factoring Special Cases Section 8-7

Goals Goal Rubric To factor perfect-square trinomials and the difference of two squares. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Perfect-Square Trinomial Difference of Two Squares

Perfect Square Trinomial

Perfect Square Trinomials Factor the polynomial 25x 2 + 20x + 4. The result is (5x + 2)2, an example of a binomial squared. Any trinomial that factors into a single binomial squared is called a perfect square trinomial.

A perfect square trinomial results after squaring a binomial Example: (2x – 5)2 Multiply it out using FOIL (2x – 5)2 = (2x – 5) (2x – 5) = 4x2 – 10x – 10x + 25 F O I L = 4x2 – 20x + 25 The first and last terms are perfect squares. The middle term is double the product of the square roots of the first and last terms.

Perfect Square Trinomials (a + b)2 = a 2 + 2ab + b 2 (a – b)2 = a 2 – 2ab + b 2 So if the first and last terms of our polynomial to be factored can be written as expressions squared, and the middle term of our polynomial is twice the product of those two expressions, then we can use these two previous equations to easily factor the polynomial. a 2 + 2ab + b 2 = (a + b)2 a 2 – 2ab + b 2 = (a – b)2

So how do we factor a Prefect Square Trinomial When you have to factor a perfect square trinomial, the patterns make it easier Product Doubled Example: Factor 36x2 + 60x + 25 Perfect Square 6x 30x 5 Perfect Square First you have to recognize that it’s a perfect square trinomial Square Root Product Square Root And so, the trinomial factors as: Check: (6x + 5)2

Example – First verify it is a Perfect Square Trinomial = (3w + 4)2 = (4x – 9)2 = (5h – 16)(5h – 4) = (a + 3)2 m2 – 4m + 4 9w2 + 24w + 16 16x2 – 72x + 81 25h2 – 100h + 64 a2 + 6a + 9 m 2m 2 3w 12w 4 4x 36x 9 Not a perfect square trinomial! 5h 40h 8 a 3a 3

Your Turn: = (m – 3)2 m2 – 6m + 9 = (2w + 7)2 4w2 + 28w + 49 = (9x – 1)2 = (3a + 2)2 m2 – 6m + 9 4w2 + 28w + 49 81x2 – 18x + 1 9a2 + 12a + 4 m 3m 3 2w 14w 7 9x 9x 1 3a 6a 2

Review: Perfect-Square Trinomial

Difference of Two Squares D.O.T.S.

Conjugate Pairs (3x + 6) (3x - 6) (r - 5) (r + 5) (2b - 1) (2b + 1) The following pairs of binomials are called conjugates. Notice that they all have the same terms, only the sign between them is different. (3x + 6) (3x - 6) and (r - 5) (r + 5) and (2b - 1) (2b + 1) and (x2 + 5) (x2 - 5) and

Multiplying Conjugates When we multiply any conjugate pairs, the middle terms always cancel and we end up with a binomial. (3x + 6)(3x - 6) = 9x2 - 36 (r - 5)(r + 5) = r2 - 25 (2b - 1)(2b + 1) = 4b2 - 1

Only TWO terms (a binomial) Difference of Two Squares Binomials that look like this are called a Difference of Squares: Only TWO terms (a binomial) 9x2 - 36 A MINUS between! The first term is a Perfect Square! The second term is a Perfect Square!

Difference of Two Squares A binomial is the difference of two square if both terms are squares and the signs of the terms are different. 9x 2 – 25y 2 – c 4 + d 4

Factoring the Difference of Two Squares A Difference of Squares! A Conjugate Pair!

Difference of Two Squares Example: Factor the polynomial x 2 – 9. The first term is a square and the last term, 9, can be written as 32. The signs of each term are different, so we have the difference of two squares Therefore x 2 – 9 = (x – 3)(x + 3). Note: You can use FOIL method to verify that the factorization for the polynomial is accurate.

Difference of Two Squares Example: Factor x2 - 64 = (x + 8)(x - 8) x2 = x • x 64 = 8 • 8 Example: Factor 9t2 - 25 = (3t + 5)(3t - 5) 9t2 = 3t • 3t 25 = 5 • 5

Difference of Two Squares Example: Factor x 2 – 16. Since this polynomial can be written as x 2 – 42, x 2 – 16 = (x – 4)(x + 4). Factor 9x2 – 4. Since this polynomial can be written as (3x)2 – 22, 9x 2 – 4 = (3x – 2)(3x + 2). Factor 16x 2 – 9y 2. Since this polynomial can be written as (4x)2 – (3y)2, 16x 2 – 9y 2 = (4x – 3y)(4x + 3y).

A Sum of Squares? A Sum of Squares, like x2 + 64, can NOT be factored! It is a PRIME polynomial.

Difference of Two Squares Example: Factor x 8 – y 6. Since this polynomial can be written as (x 4)2 – (y 3)2, x 8 – y 6 = (x 4 – y 3)(x 4 + y 3). Factor x2 + 4. Oops, this is the sum of squares, not the difference of squares, so it can’t be factored. This polynomial is a prime polynomial.

Difference of Two Squares Example: Factor 36x 2 – 64. Remember that you should always factor out any common factors, if they exist, before you start any other technique. Factor out the GCF. 36x 2 – 64 = 4(9x 2 – 16) Since the polynomial can be written as (3x)2 – (4)2, (9x 2 – 16) = (3x – 4)(3x + 4). So our final result is 36x 2 – 64 = 4(3x – 4)(3x + 4).

Recognizing D.O.T.S. Reading Math Recognize a difference of two squares: the coefficients of variable terms are perfect squares, powers on variable terms are even, and constants are perfect squares. Reading Math

Your Turn: Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 3p 2 – 9q 4 3p 2 – 9q 4 3q 2  3q 2 3p2 is not a perfect square. 3p 2 – 9q 4 is not the difference of two squares because 3p 2 is not a perfect square.

Your Turn: Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 100x 2 – 4y 2 100x 2 – 4y 2 2y 2y  10x 10x The polynomial is a difference of two squares. (10x)2 – (2y)2 a = 10x, b = 2y (10x + 2y)(10x – 2y) Write the polynomial as (a + b)(a – b). 100x2 – 4y2 = (10x + 2y)(10x – 2y)

Your Turn: Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. x 4 – 25y 6 x 4 – 25y 6 5y 3 5y 3  x 2 x 2 The polynomial is a difference of two squares. (x 2)2 – (5y 3)2 a = x2, b = 5y3 Write the polynomial as (a + b)(a – b). (x 2 + 5y 3)(x 2 – 5y 3) x 4 – 25y 6 = (x 2 + 5y 3)(x 2 – 5y 3)

Your Turn: Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 1 – 4x 2 1 – 4x 2 2x 2x  1 1 The polynomial is a difference of two squares. (1) – (2x)2 a = 1, b = 2x (1 + 2x)(1 – 2x) Write the polynomial as (a + b)(a – b). 1 – 4x 2 = (1 + 2x)(1 – 2x)

Your Turn: Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. p 8 – 49q 6 p 8 – 49q 6 7q 3 7q 3 ● p 4 p 4 The polynomial is a difference of two squares. (p 4)2 – (7q 3)2 a = p4, b = 7q3 (p 4 + 7q 3)(p 4 – 7q 3) Write the polynomial as (a + b)(a – b). p 8 – 49q 6 = (p 4 + 7q 3)(p 4 – 7q 3)

Your Turn: Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 16x 2 – 4y 5 16x 2 – 4y 5 4x  4x 4y5 is not a perfect square. 16x 2 – 4y 5 is not the difference of two squares because 4y 5 is not a perfect square.

Review: D.O.T.S.

Assignment 8-7 Exercises Pg. 531 - 532: #10 – 40 even, 48, 52