EXAMPLE: What are the possible rational zeros of P(x) = 3x3 + 11x2 ­6x ­8
Descartes' Rule of Signs: for a polynomial f(x) with real coefficients:
1) The number of positive, real zeros of f is either the number of changes in sign of f
(x) or it is that number less a positive, even integer.
2) The number of negative, real zeros of f is either the number of changes in sign of f(­x) or is that number less a positive, even integer.
EXAMPLES:
a) P(x) = 2x3 + x2 ­25x + 12
b) P(x) = 4x4 ­ 12x3 ­ 3x2 + 12x ­ 7
Mrs. Rogers' Expectations:
Finding Zeros of a Polynomial:
1) List all possible Rational Zeros (p/q list)
2) (sometimes) Use Descartes' Rule of Signs
3) Graph f(x) on your calculator to try to determine a zero to
test
4) Use long or synthetic division to verify the zero algebraically and to
reduce the polynomial by one degree
5) Repeat steps 3 and 4 until the polynomial is reduced to a
quadratic
6) Solve the quadratic to find the remaining zeros
7) List all zeros in a solution set using braces { }
Graphing a Polynomial:
1) Use the Leading Term Test to determine the end behavior of the graph as
x⇒±∞
2) Find f(0) to determine the y-intercept
3) Use methods to find a zero and division to reduce the polynomial and find
all zeros (x-intercepts)
4) Use the even/odd powers of (x - c) theorem to determine the graph
behavior (crosses or bounces) at each zero (x-intercept)
5) Use n-1 at most turning points rule
6) Sketch a possible graph of the polynomial by using all of this
information
EXAMPLES: a) P(x) = 2x3 + x2 ­ 25x + 12
Find the zeros of the polynomial function. If a zero is a multiple zero, state its multiplicity.
b) P(x) = x3 ­ 8x2 + 8x + 24