4.5 Supplemental Notes - Factoring Special Products - Solutions Algebra 3 4.5 Supplemental Notes - Factoring Special Products - Solutions

Use the Patterns! (2x + 3)2 =4x² + 12x + 9 =4p² - 16 (2p - 4) (2p + 4) First and last terms are perfect squares! (2x + 3)2 =4x² + 12x + 9 Perfect Square Trinomial! The middle term is twice the product of the square roots of the first and third terms. =4p² - 16 (2p - 4) (2p + 4) Difference of two squares (DTS)! The difference of… two squares! First and last terms are perfect squares! (2x - y)2 =4x² - 4xy + y² Perfect Square Trinomial! The middle term is twice the product of the square roots of the first and third terms. The key is to recognize when you see a perfect square trinomial or a difference of squares!

Examples: 1) 𝑥 2 +10𝑥+25 a=x, b=5 = (𝑥) 2 +2 5 𝑥 + (5) 2 =(𝑥+5)(𝑥+5) = (𝑥+5) 2

Examples: 2) 9𝑥 2 +42𝑥+49 a=3x, b=7 = (3𝑥) 2 +2 7 3𝑥 + (7) 2 =(3𝑥+7)(3𝑥+7) = (3𝑥+7) 2

Examples: 3) 𝑥 2 −10𝑥+25 a=x, b=5 = (𝑥) 2 −2 5 𝑥 + 5 2 =(𝑥−5)(𝑥−5) = (𝑥−5) 2

Examples: 4) 100𝑥 2 −60𝑥+9 a=10x, b=3 = (10𝑥) 2 −2 3 10𝑥 + 3 2 =(10𝑥−3)(10𝑥−3) = (10𝑥−3) 2

Once you can see the patterns easily, you can factor in one step: 𝑥 2 +10𝑥+25 a=x, b=5 = (𝑥+5) 2 9𝑥 2 +42𝑥+49 a=3x, b=7 = (3𝑥+7) 2 𝑥 2 −10𝑥+25 a=x, b=5 = (𝑥−5) 2 100𝑥 2 −60𝑥+9 a=10x, b=3 = (10𝑥−3) 2

Factoring Patterns! a² + 2ab + b2 =(a + b)2 a² - b2 =(a - b)(a + b) First and last terms are perfect squares! a² + 2ab + b2 =(a + b)2 Perfect Square Trinomial! The middle term is twice the product of the square roots of the first and third terms. a² - b2 =(a - b)(a + b) Difference of two squares (DTS)! The difference of… two squares! First and last terms are perfect squares! a² - 2ab + b² =(a - b)2 Perfect Square Trinomial! The middle term is twice the product of the square roots of the first and third terms. The key is to recognize when you see a perfect square trinomial or a difference of squares!

Factor! 2x²- 18 =2(x + 3)(x – 3) =2(x²- 9) =(7t + ½r)(7t – ½r) Difference of Squares! =(7t + ½r)(7t – ½r) 49t²- ¼r2 Difference of Squares! 81x²- 25y² =(9x – 5y)(9x + 5y) Difference of Squares! 27x²- 12 =3(3x + 2)(3x – 2) =3(9x²- 4) Difference of Squares!

Factor! -3x²- 18x - 27 =-3(x + 3)2 =-3(x²+ 6x + 9) =(3y – 10)2 Perfect Square Trinomial! =(3y – 10)2 9y²- 60y + 100 Perfect Square Trinomial! 2x²- 12x + 18 =2(x – 3)2 =2(x²- 6x + 9) Perfect Square Trinomial! 49x²+ 84x + 36 =(7x + 6)2 Perfect Square Trinomial!

Solve! (x – 5)2 = 0 (x-5)(x-5)=0 3x²- 30x = -75 x-5=0 or x-5=0 Divide each side by 3! x²- 10x + 25 = 0 x = 5 Perfect Square Trinomial! (6y + 11)(6y – 11) = 0 6y+11=0 6y-11=0 6y=-11 6y=11 Y=-11/6 y=11/6 36y²- 121 = 0 Difference of Squares! y = {-11/6, 11/6}

Solve! -6x²+ 8x + 14 = 0 (x )(3x ) = 0 x+1=0 3x-7=0 x=-1 3x=7 x=7/3 + - 1 7 Divide each side by -2! 3x²- 4x – 7 = 0 x = {-1, 7/3}

Solve! 4x²- 1 = 0 (2x + 1)(2x – 1) = 0 2x+1=0 2x-1=0 2x=-1 2x=1 Difference of Squares! 7x²- 10x = -3 7x²- 10x + 3 = 0 x = {-½, ½} (7x )(x ) = 0 7x-3=0 x-1=0 7x=3 x=1 x=3/7 – – 3 1 x = {3/7, 1}

Solve! 32x²- 80x + 50 = 0 (4x – 5)2 = 0 4x-5=0 4x=5 16x²- 40x + 25 = 0 Divide each side by 2! 16x²- 40x + 25 = 0 Perfect Square Trinomial! x = 5/4

HW Find and try at least one problem that fits each of the special product factoring rules: (look in 4.4 and 4.5) a² + 2ab + b2 =(a + b)2 a² - b2 =(a - b)(a + b) a² - 2ab + b² =(a - b)2