6-3 Dividing Polynomials

Transcription

1 Polynomials can be divided using long division just like you learned with numbers. Divide) Remainder 24 6 = 5 4 6

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3 Example : Using Long Division to Divide a Polynomial Divide using long division. ( y 2 + 2y + 25) (y ) Step Write the dividend in standard form, including terms with a coefficient of 0. 2y y 2 + 0y + 25 Step 2 Write division in the same way you would when dividing numbers. y 2y y 2 + 0y + 25

4 Step Divide. Example Continued 2y 2 + 5y + 5 y 2y y 2 + 0y + 25 Notice that y times 2y 2 is 2y. Write 2y 2 above 2y. (2y 6y 2 ) Multiply y by 2y 2. Then 5y 2 + 0y subtract. Bring down the next term. Divide 5y 2 by y. (5y 2 5y) Multiply y by 5y. Then 5y + 25 subtract. Bring down the next term. Divide 5y by y. (5y 45) Multiply y by 5. Then subtract. 70 Find the remainder.

5 Example Continued Step 4 Write the final answer. y 2 + 2y + 25 y = 2y 2 + 5y y

6 Check It Out! Example a Divide using long division. (5x 2 + 8x 2) (x + ) Step Write the dividend in standard form, including terms with a coefficient of 0. 5x 2 + 8x 2 Step 2 Write division in the same way you would when dividing numbers. x + 5x 2 + 8x 2

7 Check It Out! Example a Continued Step Divide. 5x + x + 5x 2 + 8x 2 (5x 2 + 5x) x 2 (x + ) Notice that x times 5x is 5x 2. Write 5x above 5x 2. Multiply x + by 5x. Then subtract. Bring down the next term. Divide x by x. Multiply x + by. Then subtract. Find the remainder.

8 Check It Out! Example a Continued Step 4 Write the final answer. 5x 2 + 8x 2 x + = 5x + x +

9 Check It Out! Example b Divide using long division. (x 2 + 5x 28) (x ) Step Write the dividend in standard form, including terms with a coefficient of 0. x 2 + 5x 28 Step 2 Write division in the same way you would when dividing numbers. x x 2 + 5x 28

10 Check It Out! Example b Continued Step Divide. x + 8 x x 2 + 5x 28 (x 2 x) 8x 28 (8x 24) 4 Notice that x times x is x 2. Write x above x 2. Multiply x by x. Then subtract. Bring down the next term. Divide 8x by x. Multiply x by 8. Then subtract. Find the remainder.

11 Check It Out! Example b Continued Step 4 Write the final answer. x 2 + 5x 28 x = x x

12 Synthetic division is a shortcut method for dividing a polynomial by a linear binomial (x a). The divisor must be a binomial. The x must have a coefficient of. Use the opposite sign of a.

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14 Example 2A: Using Synthetic Division to Divide by a Linear Binomial Divide using synthetic division. (x 2 + 9x 2) (x ) Step Find a. Then write the coefficients and a in the synthetic division format. a = 9 2 For (x ), a =. Write the coefficients of x 2 + 9x 2.

15 Example 2A Continued Step 2 Bring down the first coefficient. Then multiply and add for each column Step Write the quotient. x x Draw a box around the remainder,.

16 Check Multiply (x ) Example 2A Continued x (x ) + 0 (x ) + = x 2 + 9x 2 x x x (x )

17 Example 2B: Using Synthetic Division to Divide by a Linear Binomial Divide using synthetic division. (x 4 x + 5x ) (x + 2) Step Find a. a = 2 For (x + 2), a = 2. Step 2 Write the coefficients and a in the synthetic division format Use 0 for the coefficient of x 2.

18 Example 2B Continued Step Bring down the first coefficient. Then multiply and add for each column Draw a box around the remainder, 45. Step 4 Write the quotient. x 7x 2 + 4x x + 2 Write the remainder over the divisor.

19 Check It Out! Example 2a Divide using synthetic division. (6x 2 5x 6) (x + ) Step Find a. a = For (x + ), a =. Step 2 Write the coefficients and a in the synthetic division format Write the coefficients of 6x 2 5x 6.

20 Check It Out! Example 2a Continued Step Bring down the first coefficient. Then multiply and add for each column Draw a box around the remainder, 6. Step 4 Write the quotient. 6x x + Write the remainder over the divisor.

21 Check It Out! Example 2b Divide using synthetic division. (x 2 x 8) (x 6) Step Find a. a = 6 For (x 6), a = 6. Step 2 Write the coefficients and a in the synthetic division format. 6 8 Write the coefficients of x 2 x 8.

22 Check It Out! Example 2b Continued Step Bring down the first coefficient. Then multiply and add for each column There is no remainder. Step 4 Write the quotient. x +

23 Homework pg. 426 # s 9-29 odd, 7-4 odd