1. Polynomial Division Division Long and Short

2. 8/10/2013 Polynomial Division 2 Division of Polynomials Long Division (by hand): Polynomial Functions 245 ) 9815 4 980 15 0 Quotient Remainder 9815 245 We just did ! So, = 40+ 3 49  Can we do this with polynomials ? = 40 + 15 245 * * See Notes page

3. 8/10/2013 Polynomial Division 3 General Division of Polynomials For polynomials f(x) and d(x), d(x) ≠ 0 where Q(x) is the quotient, r(x) is the remainder, d(x) is the divisor and f(x) is the dividend Thus where either r(x) = 0 or deg r(x) < deg d(x) Polynomial Functions + f(x) d(x) = Q(x) r(x) d(x) f(x) = d(x) ∙ Q(x) + r(x)

4. 8/10/2013 Polynomial Division 4 Algebraic Monomial Division Example: Polynomial Functions 12x 4 + 27x 3 – 9x 2 + 6x – 2 3x 2 Quotient Remainder = 12x 4 3x 2 + – + – 9x 2 3x 2 6x 3x 2 27x 3 3x 2 2 2 – 2 x + = 4x 2 + 9x 3 – 3x 2 2 – 6x + = 4x 2 + 9x 3 –

5. 8/10/2013 Polynomial Division 5 Arithmetic Monomial Division Example Polynomial Functions 12x 4 + 27x 3 – 9x 2 + 6x – 2 3x 2 ) 4x 2 12x 4 27x 3 + 9x 27x 3 – 9x 2 – 3 – 9x 2 6x – 2 Remainder Quotient Note: deg r(x) = 1 < deg d(x) = 2

6. 8/10/2013 Polynomial Division 6 Division by Linear Binomials Example Polynomial Functions x 4 + 3x 3 – 4x + 1 x + 2 x3x3 x 4 + 2x 3 x3x3 + x 2 x 3 + 2x 2 – 2x 2 – 2x – 2x 2 – 4x 1 – 4x ) Quotient Remainder Question: Can we do this faster or more simply ?

7. 8/10/2013 Polynomial Division 7 Arithmetic operations involve only the coefficients Synthetic Division 2x 4 – 3x 3 + 5x 2 + 4x + 3 x + 2 ) 2x 3 2x 4 + 4x 3 –7x 3 – 7x 2 –7x 3 – 14x 2 19x 2 – 34 19x 2 + 38x – 34x + 4x +19x + 5x 2 + 3 – 34x – 68 71 Remainder  Example: NOTE: deg r(x) = 0 < 1 = deg d(x)

8. 8/10/2013 Polynomial Division 8  Using synthetic division we deal only with the coefficients Synthetic Division 2x 4 – 3x 3 + 5x 2 + 4x + 3 x + 2 )  Example: 2 –3 5 4 3 2 2 4 –7 –14 19 38 –34 –68 71 Subtract d(x) = x + 2 Remainder –2 2 –4 –7 14 19 –38 –34 68 71 Add d(x) = x – (–2) 2 –3 5 4 3

9. 8/10/2013 Polynomial Division 9 Degree Facts For any polynomials A(x), B(x) deg (A(x) B(x)) deg (A(x) + B(x)) Examples: deg ( (x 2 + 1) (3x 3 – 4x 2 + 5x + 7) ) deg ( (3x 2 – 4) + (2x 4 + 6x + 3) ) Polynomial Functions = deg A(x) + deg B(x) = max { deg A(x), deg B(x) } = 2 + 3 = 5 = max { 2, 4 } = 4

10. 8/10/2013 Polynomial Division 10 Division Algorithm for Polynomials Consider polynomial functions f(x), d(x) with 0 < deg d(x) < deg f(x) There exist unique polynomial functions Q(x) and r(x) such that f(x) = d(x) Q(x) + r(x) where either r(x) = 0 or deg r(x) < deg d(x) Polynomial Functions

11. 8/10/2013 Polynomial Division 11 Division Algorithm f(x) = d(x) Q(x) + r(x) with r(x) = 0 or deg r(x) < deg d(x) Polynomial Functions (continued) Note: This just says that f(x) d(x) Q(x) = + r(x) d(x)

12. 8/10/2013 Polynomial Division 12 Division Algorithm Polynomial Functions (continued) Example: x 2 – 1 x – 1 = x + 1 deg f(x) = 2 deg d(x) = 1, r(x) = 0 So, f(x) = d(x) Q(x) + r(x) x 2 – 1 = (x – 1) (x + 1) + 0 f(x) d(x) Q(x) = + r(x) d(x) becomes

13. 8/10/2013 Polynomial Division 13 Division Algorithm and Degrees Given f(x) = d(x) Q(x) + r(x) where either r(x) = 0 or deg r(x) < deg d(x) Polynomial Functions

14. 8/10/2013 Polynomial Division 14 Division Algorithm and Degrees Given f(x) = d(x) Q(x) + r(x) Question: Suppose deg d(x) = m > 0, deg Q(x) = n, deg f(x) = p > m What is the relationship among m, n and p ? Polynomial Functions

15. 8/10/2013 Polynomial Division 15 Division Algorithm and Degrees f(x) = d(x) Q(x) + r(x) Polynomial Functions p = deg f(x) = deg ( d(x) Q(x) + r(x) ) = max { deg (d(x) Q(x)), deg r(x) } = max { (deg d(x) + deg Q(x)), deg r(x) }

16. 8/10/2013 Polynomial Division 16 Division Algorithm and Degrees Polynomial Functions = m + n Since deg r(x) < deg d(x) then p = deg f(x) = max { (deg d(x) + deg Q(x)), deg r(x) } = max { (m + n), deg r(x) } = m < m + n max { (m + n), deg r(x) }

17. 8/10/2013 Polynomial Division 17 Division Algorithm and Degrees f(x) = d(x) Q(x) + r(x) Polynomial Functions = m + n p = deg f(x) = max { (m + n), deg r(x) } p = m + n So

18. 8/10/2013 Polynomial Division 18 Think about it !