Section+2.4

Section 2.4- Real Zeroes of Polynomial Functions Notes for October 30th Long division: __32781__ 83 __00394+(79/83)__ 83 32781 -__249__ 788 __-747__ 411 -__332__ 79 Dividing Polynomials: Put the polynomials in descending order, filling in a zero for any missing terms. Example:

2x^4-x^3-2 2x^2+x+1 __x^2-x+(x-2/2x^2+x+1)__ 2x^2+x+1 2x^4-x^3+0x^2+0x-2 __2x^4+x^3+x^2__ -2x^3-x^2+0x __-2x^3-x^2-x__ x-2 Synthetic Division: -Only works when divisor can be written as “x-k” -Put the polynomials in descending order, filling it a zero for any missing terms. Example: __2x^3-3x^2-5x-12__ X+1
 * This degree must be less than the divisor’s degree

-1 2 -3 -5 -12 The answer is 2x^2-5x-(12/x+1) __-2 5 0__ 2 -5 0 -12

Remainder Theorem: If a polynomial f(x) is divided by x-k, then the remainder is f(k). Factor Theorem: A polynomial function f(x) has a factor x-k, if and only if f(k)=0. Example: Given f(x)=x^3-2x^2+5x-1 Find f(-1)

-1 1 -2 5 -1 __-1 3 -8__ 1 -3 8 **-9** f(-1)=-9 Is x+4 a factor of 3x^2+7x-20?

-4 3 7 -20 __-12 20__ 3 -5 0 ß Yes, it is a factor!! Factored equation: (x+4)(3x-5)

Notes for November 3rd Rational Zeroes Theorem: The possible rational zeroes come from the ration of +or- p/q. Where p is the factor of the constant term, and q is the factors of the leading coefficient. Example: Find the rational zeroes F(x)=x^3-3x^2+1 P=1 Q=1 P/q= +or- 1 Test if f(x)=0

1 1 -3 0 1 __1 -2 -2__ 1 -2 -2 -1 F(1)=-1 ß This means that 1 is not a zero Test: f(-1)=0

-1 1 -3 0 1 __-1 4 -4__ 1 -4 4 -3 F(-1)=-3 ß Remainder does not work, the zeroes are either irrational or imaginary Example: Find the rational zeroes G(x)=3x^3+4x^2-5x-2 p=2,1 q=3,1 p/q=+or-:1,2,1/3,2/3 Graph to determine what would be useful.

1 3 4 -5 -2 __3 7 2__ 3 7 2 0 3x^2+7x+2=0 (3x+1)(x+2)=0 X=-1/3, x=-2, x=1 Homework for 10/30/08: Page 223 #1-23 odd Homework for 11/03/08: Page 224 #25-29 odd, #49-55 odd, #62