Multiplication: Special Cases Chapter 4.5

Sum x Difference = Difference of Two Squares (a + b)(a – b) = (a – b)(a + b) =a 2 – b 2

( 36x x – 49 – 42x 36x 2 FirstOuter Inner Last – Multiply. Use FOIL to multiply. Combine like terms. 6x + 7 )( 6x – 7 )

(6x) 2 – (7) 2 36x 2 – 49 ( 1. Multiply. 6x + 7 )( 6x – 7 ) Special Case (a + b)(a – b) = a 2 – b 2 Square the first term. Subtract the square of the second term.

(3x) 2 – (5y) 2 9x 2 – 25y 2 ( 2. Multiply. 3x – 5y )( 3x + 5y ) Special Case (a + b)(a – b) = a 2 – b 2 Square the first term. Subtract the square of the second term.

(10) 2 – (4a) – 16a 2 ( extra a )( 10 – 4a ) Special Case (a + b)(a – b) = a 2 – b 2 Square the first term. Subtract the square of the second term.

(a + b)(a – b) = (a – b)(a + b) =a 2 – b 2 Sum x Difference = Difference of Two Squares

Chapter 4.5 Multiplication: Special Cases

(a + b) 2 = (a – b) 2 =a 2 – 2ab + b 2 a 2 + 2ab + b 2 Binomial Squared = Perfect Square Trinomial

( 16a 2 – 36ab+ 81b 2 – 36ab 16a 2 FirstOuter Inner Last – 72ab+ 81b 2 3a. Multiply. Use FOIL to multiply. Combine like terms. 4a – 9b ) 4a – 9b ) (4a – 9b) ( 2

(4a) 2 – 2(4a)(9b) 16a 2 ( 3a. Multiply. 4a – 9b )2)2 Special Case (a – b) 2 = a 2 – 2ab + b 2 Square the first term. Subtract 2 times the first and second terms. Add the square of the second term. + ( - 9b) 2 – 72ab + 81b 2

(5x) 2 + 2(5x)(4) 25x 2 ( 3b. Multiply. 5x + 4 )2)2 Special Case (a + b) 2 = a 2 + 2ab + b 2 Square the first term. Add 2 times the first and second terms. Add the square of the second term. + (4) x + 16

(3x) 2 – 2(3x)(8) 9x 2 ( extra 3x – 8 )2)2 Special Case (a – b) 2 = a 2 – 2ab + b 2 Square the first term. Subtract 2 times the first and second terms. Add the square of the second term. + ( - 8) 2 – 48x + 64

(7x) 2 + 2(7x)(1) 49x 2 ( extra 7x + 1 )2)2 Special Case (a – b) 2 = a 2 – 2ab + b 2 Square the first term. Add 2 times the first and second terms. Add the square of the second term. + (1) x + 1

Binomial Squared = Perfect Square Trinomial (a + b) 2 = (a – b) 2 =a 2 – 2ab + b 2 a 2 + 2ab + b 2

Chapter 4.5 Multiplication: Special Cases

Multiplying Two Trinomials Multiplying Three Binomials

4x 3 – 2x 4x 5 + 4x 2 – 8x 3 + 3x 2 – 6x x 4 + x 3 – 2x 4 + 4x 5 – 2x 2 + x x 2 + 3x – x 4 – 13x 3 + 7x 2 – 2x 4. Multiply vertically. (4x 3 – 2x 2 + x)(x 2 + 3x – 2) Multiply each term. Combine.

( )( ) 2x 2 – 12 2x 4 – 20x– 8x 2 – 9x– 15x 2 – 6x 3 + 3x 2 + 5x 3 2x 4 + 5x + 3 x2x2 – 3x – 4 – x 3 – 20x 2 – 29x – Multiply. Multiply each term. Combine.

( – 4 (3x – 2) ( ( 18x 3 – 8x– x 2 ( (2x + 3) 6. Multiply. 9x 2 ) 3x– 2) 3x + 2 ) Sum and difference, rewrite. Special case (a + b)(a – b) = a 2 – b 2. Use FOIL. Can’t combine. (2x + 3) (3x + 2) 2x + 3 )

Multiplication: Special Cases Chapter 4.5