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Dividing Polynomials March 07, 2014 Basic Method for Solving polynomials of the type: ax3 + bx2 + cx + d = 0 & ax4 + bx3 + cx2 + dx + e = 0 1. find linear factors using graphing calculator/Theorems 2. Divide polynomial by linear factor using: 3. solve remaining quadratic long division or synthetic division 1 Dividing Polynomials March 07, 2014 Using Division to find Factors Long division a. similar to long division of 2 & 3 digit numbers b. must include "0 terms" for missing terms c. factors are the divisor & the quotient if remainder = 0 d. subtraction is used in long division 2 Dividing Polynomials March 07, 2014 Synthetic division a. simplified method of dividing polynomials which omits the variables & exponents b. the divisor number 'a' comes from (x a) (think of it as the value of x from x a = 0) reverse sign of 'a' c. must include "0" for coefficient of missing terms d. Addition is used throughout this process 3 Dividing Polynomials March 07, 2014 Divide by Linear (x - using long division Pull Pull divisor 1. divide 1 "Think 2. Align t Pull 3. multip Pull 4. bring d Pull 1. Pull Dividing a polynomial by a linear factor Rema 4 Dividing Polynomials March 07, 2014 Divide Linear factor (x - a) using long division by divisor Pull Pull Pull Subtract (changes to add & changes signs 1. divide 1st term by "x" in (x - a) "Think: what times "x" is "x2"? 2. Align terms on top by degree 3. multiply, then subtract each piece Pull 4. bring down next term, & repeat Pull 1. Pull Dividing a polynomial by a linear factor Remainder 5 Dividing Polynomials March 07, 2014 Dividing a polynomial by a linear factor using Divide by Linea (x using long division divisor Pull 2. Pull LONG DIVISION Rema 6 Dividing Polynomials March 07, 2014 Dividing a polynomial by a linear factor using Divide by Linear factor (x - a) using long division divisor Pull 2. Pull LONG DIVISION Remainder 7 Dividing Polynomials March 07, 2014 3. must be standard form, if not, include 0x2, 0x... for any missing terms 4. 8 Dividing Polynomials March 07, 2014 3. must be standard form, if not, include 0x2, 0x... for any missing terms 4. 9 Dividing Polynomials March 07, 2014 SYNTHETIC DIVISION Linear factor (x - a) *use the number part only *reverse the sign, to create the "zero" For the Pull Pull For the Polynomial *use only coefficients *include signs *fill in missing terms with "0's" 10 Dividing Polynomials 1. March 07, 2014 Use synthetic division to divide by Pull 1. write & reve Pull 2. write use " Pull 3. Bring Pull 4. Multi Pull 5. repe mult e Active Math 11 Dividing Polynomials 1. March 07, 2014 Use synthetic division to divide by Pull 1. write number from (x - a) & reverse sign in Pull 2. write the coefficients use "0" for missing terms Pull 3. Bring down 1st coefficient Pull 4. Multiply then add did not divide evenly but we do have a function value Pull 5. repeat with new number multiply & add see Active Math 12 Dividing Polynomials March 07, 2014 Divide using synthetic division 2. 3. 4. 13 Dividing Polynomials March 07, 2014 Divide using synthetic division 2. X+1 is factor, x = -1 is zero Divides Evenly x2 + 6x + 9 Depressed Polynomial 3. 1 degree less than original polynomial Does not Divide Evenly 4. 14 Dividing Polynomials March 07, 2014 Example for finding a function value using synthetic division & the Remainder Theorem Use synthetic division to find P(2) 15