Graphs of polynomials a n > 0a n < 0 n even n odd f(x)=x 3 +2x 2 -x-2g(x)=-x 4 +5x 2 -4 degree 3degree 4 degree n h(x)=a n x n +a n-1 x n-1 +…+a 1 x+a 0 2 turning points3 turning points At most (n-1) turning points

When you zoom out, the effect of the lower degree terms is difficult to see f(x)=x 3 +2x 2 -x-2 Close upZoom out looks like graph of cubic m(x)=x 3

Theorem for approximating zeros of polynomials Given a polynomial P(x)=a n x n +a n-1 x n-1 +…+a 1 x+a 0 If r is a zero of the polynomial then |r | < Max {|a n-1 /a n |, … |a 1 /a n |, |a 0 /a n |}

Using the theorem to select a graphing window Polynomial: f(x)=-2x 3 -3x 2 +9x+10 Max{3/2, 9/2, 10/2} = 5 so for any zero r, |r |<5 Thus a zero of this function must lie between -5 and 5. View graph between the x- values -5 and 5