Lesson 3.3: Solving Polynomial Equations
To solve a polynomial equation, put all of the terms on one side, and then factor the polynomial using
the
theorems, followed by
division.
e.g. a) Solve: x3  15 x  7 x 2  9
b) Use a graphing calculator to check your solutions, and to find the coordinates of the
local maximum and the local minimum of the polynomial function.
In the factorization of a polynomial function, if  x  r  is a factor that occurs m times, then we say that
r is a root with a
of m . When r is a root with odd multiplicity, the graph will
the x-axis at  r,0 . However, when r is a root with even multiplicity, the graph will
the x-axis at  r,0 .
The real number r is a zero of f  x  if and only if all of the following are true:

r is a solution, or root, of f  x   0

r is a factor of the expression that defines f (that is, f  r   0 )

When the expression that defines f is divided by x  r , the remainder is 0

r is an x-intercept of the graph of f
e.g. c) Given the following graph of the polynomial function p  x   ax3  bx2  cx  d , find
the values of a , b , c , and d .
Recall that all polynomial functions are continuous. Hence, if P is a
polynomial function and P  x1  and P  x2  have opposite signs, then
there is a real number r between x1 and x2 that is a zero of P , that is,
P  r   0 . This is known as the
.
e.g. d) Use the Location Principle to prove that the polynomial function p  x   x3  4 x  2
has an x-intercept somewhere between 1 and 2.
If P  x  is a polynomial function with real coefficients, and a  bi (where b  0 ) is a root of P  x   0 ,
then a  bi is also a root of P  x   0 . This is known as the
.
e.g. e) The zeros of a cubic polynomial function P  x  include 1 and 3  4i . Also, P  0  50 .
Write the function P  x  in standard form.