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6.4 Multiplying/Dividing Polynomials 2/8/2013

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Example 1 Multiply Polynomials Vertically Find the product. () x 2x 2 4x4x7 – + () 2x – SOLUTION Line up like terms vertically. Then multiply as shown below. x 2x 2 4x4x7 – + 2x – × 2x 22x 2 8x8x+14 –– Multiply by 2. x 2x 2 4x4x7 – + – x 3x 3 7x7x+ – 4x 24x 2 Multiply by x. x 2x 2 4x4x7 – + x 3x 3 15x+ – 2x 22x 2 +14 Combine like terms.

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Example 2 Multiply Polynomials Horizontally Find the product. () 4+3x3x () 5x 25x 2 x6 – + a. 4+3x3x () 5x 25x 2 x6 – + () 5x 25x 2 x6 – + = Use distributive property. SOLUTION () 4+3x3x () 5x 25x 2 x6 – + a. + 15x 3 18x – +20x 2 4x4x24 – + = 3x 23x 2 Use distributive property. 15x 3 +24 – + = 20x 2 3x 23x 2 () + 18x4x4x – + () Group like terms. 15x 3 14x – + = 23x 2 24 – Combine like terms.

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Example 2 Multiply Polynomials Horizontally () 2x – () 1x – () 3x + To multiply three polynomials, first multiply two of the polynomials. Then multiply the result by the third polynomial. () 2x – () x 2x 2 2x2x3 – + = Multiply () 1x – ( )3x +. b. () 2x – () 1x – () 3x + + x 3x 3 3x3x – +2x 22x 2 4x4x6 – = 2x 22x 2 – Use distributive property. x 3x 3 +6+ = 2x 22x 2 2x 22x 2 () 3x3x4x4x – () –– + Group like terms. + x 3x 3 7x7x – 6 = Combine like terms. 2x – () x 2x 2 2x2x3 – + = () x 2x 2 2x2x3 – + Use distributive property.

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Checkpoint Multiply Polynomials Find the product. Use either a horizontal or vertical format. 1. () 1+x () x 2x 2 x2++ 2. () 2x 22x 2 x4 – + () 3x – ANSWER 2x 22x 2 3x3x2x 3x 3 +++ 7x 27x 2 7x7x122x 32x 3 + ––

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Checkpoint Multiply Polynomials Find the product. Use either a horizontal or vertical format. 3. () 1+2x2x () 3x 23x 2 x1 – + ANSWER 5x 25x 2 x16x 36x 3 + –– x 2x 2 10x8x 3x 3 + – + () 1x – 4. () 4x + () 2x –

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Checkpoint Use Special Product Patterns Find the product. () 7z – () 7z + 5. ANSWER z 2z 2 49 – 6. ()2)2 23y3y + 12y9y 29y 2 4++ ANSWER 64x 3 + – 12x48x 2 – 1 ANSWER 7. – ()3)3 14x4x

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Example 4 Use Long Division Find the quotient 985 23. ÷ Divide 98 by 23. 985 23 -92 Subtract the product. 4 () 23 = 92 65 Bring down 5. Divide 65 by 23. 19 Remainder ANSWER The result is written as. 23 19 42 -46 Subtract the product. 2 () 23 = 46 42

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Example 5 Use Polynomial Long Division Find the quotient. () 4x + x 3x 3 + – 6x6x3x 23x 2 – 4 () ÷ Rewrite in standard form. () 4x + x 3x 3 + – 6x6x3x 23x 2 – 4 () ÷ Write division in the same format you use to divide whole numbers. x 3x 3 +4x 24x 2 Subtract the product. () 4x + x 2x 2 = x 3x 3 4x 24x 2 + – 6x6xx 2x 2 – Bring down - 6x. Divide –x 2 by x – 4x4xx 2x 2 – Subtract the product. () 4x + x = x 2x 2 4x4x ––– – 2x2x – 4 Bring down - 4. Divide -2x by x 4 Remainder x 3x 3 + – 6x6x3x 23x 2 – 4x+4 x 3x 3 ÷x = x 2x 2 ANSWER The result is written as. x 2x 2 –– x2 x+4 4 + – 2x2x – 8 Subtract the product () 4x + 2 = 2x2x8 –––. x2x2 -x -2 - +

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Example 5 Use Polynomial Long Division CHECKYou can check the result of a division problem by multiplying the divisor by the quotient and adding the remainder. The result should be the dividend. +4x 2x 2 –– x2 () x+4 () = x+4 () x 2x 2 – xx+4 () – 2x+4 () +4 = x 3x 3 –– 4x4x+44x 24x 2 x 2x 2 + – 2x2x – 8 = x 3x 3 – 6x6x3x 23x 2 + – 4

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Checkpoint Use Long Division Use long division to find the quotient. 8. 5x5x4x 34x 3 + () +1x ( +1 ) ÷ ANSWER 4x4x4x24x2 + – 9+ x+1 8 –

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Homework: 6.4 p.318 #15-51 (x3), 81-83

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