1. Dividing by a Monomial Write the division as a fraction and use the quotient of powers property. When dividing polynomials, you can check your work using multiplication.
2. Example 1 Divide monomialsDivide –8x5 by 2x2.SOLUTIONWrite the division as a fraction and use the quotientof powers property. –8x5 –8x5 ÷ (2x2) = Write as fraction. 2x2 – 8 x5 Rewrite using product rule for = • 2 fractions. 2 x –8 = • x5 – 2 Quotient of powers property 2
3. Example 1 Divide monomials = –4x3 Simplify.
4. Example 2 Multiple Choice Practice4x3 8 =16x 1 x5 1 4x5 4x5 12 12x54x3 4 x3 8 = • 8 Rewrite using product rule for fractions.16x 16 x 1 = • x–5 Quotient of powers property 4 1 1 = • 5 Definition of negative exponents 4 x
5. Example 2 Multiple Choice Practice 1 Simplify. = 5 4xANSWER The correct answer is B.
6. Example 3 Divide a polynomial by a monomialDivide 4x3 + 8x2 + 10x by 2x.SOLUTION 4x3 + 8x2 + 10x( 4x3 + 8x2 + 10x ) ÷ 2x = Write as fraction. 2x 4x3 8x2 10x = + + Divide each term 2x 2x 2x by 2x. = 2x2 + 4x + 5 Simplify.
7. Example 3 Divide a polynomial by a monomialCHECK Check to see if the product of 2x and 2x2 + 4x + 5 is 4x3 + 8x2 + 10x. ? 2x ( 2x2 + 4x + 5) = 4x3 + 8x2 + 10x ? 2x ( 2x2) + 2x (4x ) + 2x (5 ) = 4x3 + 8x2 + 10x 4x3 + 8x2 + 10x = 4x3 + 8x2 + 10x
8. Division with Algebra Tiles Pg. 540
9. Dividing by a Binomial To divide a polynomial by a binomial, use long division.
10. Example 4 Divide a polynomial by a binomialDivide x2 + 2x – 3 by x – 1.SOLUTIONSTEP 1 Divide the first term of x2 + 2x – 3 by the first term of x – 1. x x – 1 x2 + 2x – 3 Think: x2 ÷ x = ? x2 – x Multiply x – 1 by x. 3x Subtract x2 – x from x2 + 2x.
11. Example 4 Divide a polynomial by a binomialSTEP 2 Bring down –3. Then divide the first term of 3x – 3 by the first term of x – 1. x + 3 x – 1 x2 + 2x – 3 x2 – x 3x – 3 Think: 3x ÷ x = ? 3x – 3 Multiply x – 1 by 3. 0 Subtract 3x – 3 from 3x – 3; remainder is 0.ANSWER ( x2 + 2x – 3) ÷ (x – 1) = x + 3
12. Nonzero Remainders When you obtain a nonzero remainder, apply the following rule: Re mainder Dividend Divisor Quotient Divisor 2 2 5 3 1 Which is really 1 3 3 2 12(2 x 11x 9) (2 x 3) x 7 2x 3
13. Example 5 Divide a polynomial by a binomialDivide 2x2 + 11x – 9 by 2x – 3. x + 7 2x – 3 2x2 + 11x – 9 2x2 – 3x Multiply 2x – 3 by x. 14x – 9 Subtract 2x2 – 3x. Bring down – 9. 14x – 21 Multiply 2x – 3 by 7. 12 Subtract 14x – 21; remainder is 12. 12ANSWER (2x2 + 11x – 9) ÷ ( 2x – 3) = x + 7 + 2x – 3
14. Example 6 Rewrite polynomialsDivide 5y + y2 + 4 by 2 + y. y + 3 y + 2 y2 + 5y + 4 Rewrite polynomials. y2 + 2y Multiply y + 2 by y. 3y + 4 Subtract y2 + 2y. Bring down 4. 3y + 6 Multiply y + 2 by 3. –2 Subtract 3y + 6; remainder is – 2. –2ANSWER (5y + y2 + 4) ÷ ( 2 + y) = y + 3 + y +2
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