6.4 Multiplying/Dividing Polynomials

1. 6.4 Multiplying/Dividing Polynomials 2/8/2013

2. Example 1 SOLUTION Line up like terms vertically. Then multiply as shown below. Multiply by 2. – – x2 + 4x 7 – x3 + 4x2 7x × – x 2 ( ) ( ) – – x2 + 4x 7 x 2 Multiply by x. – – 2x2 8x + 14 – – x2 x2 + + 4x 4x 7 7 – x3 + 2x2 15x + 14 Combine like terms. Multiply Polynomials Vertically Find the product .

3. Example 2 ( ) 3x + 4 Combine like terms. ( ) – 5x2 + x 6 SOLUTION a. ( ) ( ) – 3x + 4 5x2 + x 6 Use distributive property. 3x + 4 = ( ( ) ) – – 5x2 5x2 + + x x 6 6 – – + 15x3 + 3x2 18x 20x2 + 4x 24 = Use distributive property. ( ) ( ) – – + 15x3 + 3x2 20x2 + 24 18x + 4x = – – 15x3 + 23x2 14x 24 = Multiply Polynomials Horizontally Find the product. a. Group like terms.

4. Example 2 ( ) ( ) – – x 2 x 1 ( ) Group like terms. x 3 + Use distributive property. Combine like terms. – + x3 7x 6 = ( ) ( ( ) ) ( ) – – – x 3 x x 2 2 x 1 + ( ) ( – x 1 x 3 ) + = ( ( ( ) ) ) – – – x2 x2 x2 + + + 2x 2x 2x 3 3 3 – x 2 . = Multiply – – – + x3 + 2x2 3x 2x2 4x 6 = ( ) ( ) – – – x3 + 2x2 2x2 + 3x 4x + 6 = Multiply Polynomials Horizontally b. To multiply three polynomials, first multiply two of the polynomials. Then multiply the result by the third polynomial. Use distributive property.

5. Checkpoint ( ) ( ) x + 1 x2 + x + 2 x3 + 2x2 + 3x + 2 ANSWER ( ) ( ) – – x 3 2x2 x + 4 2. – – 2x3 7x2 + 7x 12 ANSWER Multiply Polynomials Find the product. Use either a horizontal or vertical format. 1.

6. Checkpoint ( ) ( ) – – x 2 x 1 ( ) x 4 + ( ) ( ) – 2x + 1 3x2 + x 1 ANSWER – – 6x3 + 5x2 x 1 4. – ANSWER x3 + x2 10x + 8 Multiply Polynomials Find the product. Use either a horizontal or vertical format. 3.

7. Checkpoint ANSWER 5. – z2 49 6. ANSWER ( )2 3y 2 + 7. ( )3 – 4x 1 ANSWER ( ) ( ) – z 7 + z 7 – – 64x3 48x2 + 12x 1 9y2 + 12y + 4 Use Special Product Patterns Find the product.

8. Example 4 Find the quotient 98523. ÷ Divide 98 by 23. 985 23 -92 Subtract the product . Subtract the product . Bring down 5. Divide 65 by 23. 65 -46 ( ( ) ) = = 2 4 23 23 46 92 Remainder 19 19 42 The result is written as . ANSWER 23 Use Long Division 2 4

9. Example 5 – – x + 4 x3 + 3x2 6x 4 x3 ÷ x x2 = Subtract the product . ( ) = x 4 x2 x3 4x2 + + x3 + 4x2 ( ) ( ) – – x 4 x3 + 3x2 6x 4 – – ÷ + x2 6x Bring down - 6x. Divide –x2by x Subtract the product . ( ) – – – = x 4 x x2 4x + ( ) ( ) – – x 4 x3 6x 4 + 3x2 + – – 2x 4 – – x2 4x – – 2x 8 . Subtract the product ( ) – – – = x 4 2 2x 8 + Bring down - 4. Divide -2x by x ANSWER 4 4 Remainder The result is written as . – – x2 x 2 + x + 4 Use Polynomial Long Division Find the quotient . ÷ Rewrite in standard form. Write division in the same format you use to divide whole numbers. x2 -2 -x - - + + + +

10. Example 5 ( ) ( ) ( ) ( ) ( ) – – – – x2 x 2 x + 4 + 4 x2 x + 4 x x + 4 2 x + 4 + 4 = – – – – x3 + 4x2 x2 4x 2x 8 + 4 = – – x3 + 3x2 6x 4 = Use Polynomial Long Division CHECK You 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.

11. Checkpoint 8. ANSWER – 8 – 4x2 4x + 9 + x + 1 ( ) ( ) 4x3 + 5x + 1 x + 1 ÷ Use Long Division Use long division to find the quotient.

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