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Understanding Special Cases in Multiplication: Difference of Two Squares and Perfect Squares

This chapter explores the special cases of multiplication, focusing on the difference of two squares and perfect square trinomials. We introduce the formula ( (a+b)(a-b) = a^2 - b^2 ) and ( (a+b)^2 = a^2 + 2ab + b^2 ). Through various examples, including multiplying binomials and trinomials, we apply techniques such as the FOIL method and vertical multiplication. This guide aims to clarify these concepts and provide practice problems for mastery in multiplication techniques.

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Understanding Special Cases in Multiplication: Difference of Two Squares and Perfect Squares

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  1. Chapter 4.5 Multiplication: Special Cases

  2. Difference of Two Squares Sum x Difference = (a + b)(a – b) = a2– b2 (a – b)(a + b) = a2– b2

  3. 1. Multiply. Outer First )( Last Inner ( + 7 6x ) 6x – 7 – 49 + 42x 36x2 – 42x 36x2 – 49 Use FOIL to multiply. Combine like terms.

  4. 1. Multiply. )( 6x – 7 ( ) 6x + 7 (7)2 – (6x)2 36x2 – 49 Special Case (a + b)(a – b) = a2 – b2 Square the first term. Subtract the square of the second term.

  5. 2. Multiply. 3x + 5y ( ) 3x – 5y )( (5y)2 – (3x)2 9x2– 25y2 Special Case (a + b)(a – b) = a2 – b2 Square the first term. Subtract the square of the second term.

  6. extra. 10 – 4a ( ) 10 + 4a )( (4a)2 – (10)2 100 – 16a2 Special Case (a + b)(a – b) = a2 – b2 Square the first term. Subtract the square of the second term.

  7. Difference of Two Squares Sum x Difference = (a + b)(a – b) = a2– b2 (a – b)(a + b) = a2– b2

  8. Chapter 4.5 Multiplication: Special Cases

  9. Perfect Square Trinomial Binomial Squared = (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2

  10. 3a. Multiply. Outer First Last Inner (4a – 9b) 2 4a ) ( – 9b ( – 9b 4a ) + 81b2 – 36ab 16a2 – 36ab 16a2 – 72ab + 81b2 Use FOIL to multiply. Combine like terms.

  11. 3a. Multiply. ( 4a – 9b )2 – (4a)2 + (-9b)2 2(4a)(9b) + 81b2 16a2 – 72ab Special Case (a – b)2 = a2 – 2ab + b2 Square the first term. Subtract 2 times the first and second terms. Add the square of the second term.

  12. 3b. Multiply. ( 5x + 4 )2 + (5x)2 + (4)2 2(5x)(4) + 16 25x2 + 40x Special Case (a + b)2 = a2 + 2ab + b2 Square the first term. Add 2 times the first and second terms. Add the square of the second term.

  13. extra ( 3x – 8 )2 – (3x)2 + (-8)2 2(3x)(8) + 64 9x2 – 48x Special Case (a – b)2 = a2 – 2ab + b2 Square the first term. Subtract 2 times the first and second terms. Add the square of the second term.

  14. extra ( 7x + 1 )2 + (7x)2 + (1)2 2(7x)(1) + 1 49x2 + 14x Special Case (a – b)2 = a2 – 2ab + b2 Square the first term. Add 2 times the first and second terms. Add the square of the second term.

  15. Perfect Square Trinomial Binomial Squared = (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2

  16. Chapter 4.5 Multiplication: Special Cases

  17. Multiplying Two Trinomials Multiplying Three Binomials

  18. 4. Multiply vertically. (4x3 – 2x2 + x)(x2 + 3x – 2) 4x3 – 2x2 + x + 3x – 2 x2 + 4x2 – 2x – 8x3 + 12x4 – 6x3 + 3x2 + 4x5 – 2x4 + x3 – 13x3 4x5 + 7x2 + 10x4 – 2x Multiply each term. Combine.

  19. 5. Multiply. 2x2 + 5x ( )( ) x2 + 3 – 3x – 4 – 20x + 3x2 – 9x – 12 – 8x2 + 5x3 – 15x2 2x4 – 6x3 – 20x2 2x4 – 29x – x3 – 12 Multiply each term. Combine.

  20. 6. Multiply. (3x – 2) (2x + 3) (3x + 2) ( ( 3x 3x ) – 2 + 2 (2x + 3) ) ) ( ( 9x2 2x + 3 4 ) – – 12 – 8x 18x3 + 27x2 Sum and difference, rewrite. Special case (a + b)(a – b) = a2 – b2. Use FOIL. Can’t combine.

  21. Chapter 4.5 Multiplication: Special Cases

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