Section 2.3 Polynomial and Synthetic Division Long Division of polynomials Ex. (6x 3 -19x 2 +16x-4) divided by (x-2) Ex. (x 3 -1) divided by (x-1) Ex (2x.

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Presentation transcript:

Section 2.3 Polynomial and Synthetic Division Long Division of polynomials Ex. (6x 3 -19x 2 +16x-4) divided by (x-2) Ex. (x 3 -1) divided by (x-1) Ex (2x 4 +4x 3 -5x 2 +3x-2) divided by (x 2 + 2x-3)

Synthetic Division Works when dividing by a binomial of the form (x-k) Use for examples on first slide: Ex. (6x 3 -19x 2 +16x-4) divided by (x-2) Ex. (x 3 -1) divided by (x-1) Ex (2x 4 +4x 3 -5x 2 +3x-2) divided by (x 2 + 2x-3) Write answer as a polynomial

The Remainder Theorem If a polynomial f(x) is divided by (x-k), then the remainder is r; r=f(k). f(x)=3x 3 +8x 2 +5x-7; what is f(-2)? f(-2)=-9, so (-2,-9) is on the graph

The factor Theorem A polynomial f(x) has a factor (x-k) iff f(k)=0. Is (x-2) a factor of f(x)=2x 4 +7x 3 -4x 2 -27x-18 ? synthetically divide the remaining polynomial Is (x+3) a factor of f(x)=2x 4 +7x 3 -4x 2 -27x-18 ? Completely factor 2x 4 +7x 3 -4x 2 -27x-18 and find the four zeros.

Using the remainder r = f(k) If r=0, then (x-k) is a factor of f(x) If r=0, then (k,0) is an x-intercept of the graph of f