3.2 Dividing Polynomials

1. 3.2 Dividing Polynomials 11/28/2012

2. Review: Quotient of Powers Ex. In general:

3. 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

4. Example 1 – – x + 4 x3 + 3x2 6x 4 x3 ÷ x x2 = Subtract the product . ( ) = x 4 x2 x3 4x2 + + x3 + 4x2 – – x2 6x Bring down - 6x. Divide –x2by x Subtract the product . ( ) – – – = x 4 x x2 4x + – – 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 Divide: x2 -2 -x - - + + + +

5. Synthetic division: Is a method of dividing polynomials by an expression of the form x - k

6. Example 1 4 ANSWER – – x2 x 2 + x + 4 Using Synthetic division Divide: x – (-4) in x – k form k add Coefficients of powers of x -4 1 3 -6 -4 -4 8 4 multiply 4 1 -2 -1 remainder coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

7. k Evaluate when x = -4: = -64 + 48 + 24 – 4 = -16 + 24 – 4 = 8 – 4 = 4 Isn’t this the remainder when we performed synthetic division? Remainder Thm: If a polynomials f(x) is divided by x – k, then the remainder is r = f(k)

8. Example 2 Using Synthetic division and Remainder Theorem Evaluate using synthetic division and Remainder Thm: k add 3 2 -7 0 6 -14 Coefficients of powers of x 6 -9 -3 -9 multiply -23 -3 2 -3 -1 remainder P(3)= -23

9. Example 3 Use Polynomial Long Division Can’t use synthetic division because it isn’t being divided by x-k Divide: = - + - = + - + - + = - + - remainder +

10. Homework: Worksheet 3.2 #1-5all, 11-19odd, 23-25all