Polynomial Functions, Zeros, Factors and Intercepts (1)

Tutorial and problems with detailled solutions on finding polynomial functions given their zeros and/or graphs and other information.

Review
Let p(x) be a polynomial function with real coefficients. If p(s) = 0

s is a zero for the polynomial function p(x).

s is a solution to the equation p(x) = 0

x - s is a factor of p(x).

The point (s , 0) is an x intercept of the graph of p(x).

Example: let p(x) = x^{3} - 2·x^{2} + 2·x - 4

p(2) = 2^{3} - 2*2^{2} + 2*2 - 4

= 8 - 8 + 4 - 4
= 0
2 is a zero of p(x).

x = 2 is a solution of p(x) = 0

p(x) can be written in factored form as
p(x) =(x - 2)·(x^{2} + 2)

The graph of p(x) below shows an x intercept at x = 2.

TUTORIAL

Example - Problem 1: The graph below is that of a polynomial function p(x) with real coefficients. The degree of p(x) is 3 and the zeros are assumed to be integers. Find p(x).

Solution to Problem 1:

The graph has 2 x intercepts: -1 and 2. The x intercept at -1 is of multiplicity 2. p(x) can be written as follows

p(x) = a(x + 1)^{2}(x - 2) , a is any real constant not equal to zero.

To find a we need to use more information in the graph. The y intercept is at (0 , -2), which means that p(0) = -2

a(0 + 1)^{2}(0 - 2) = -2

Solve the above equation for a to obtain

a = 1

p(x) is given by

p(x) = (x + 1)^{2}(x - 2)

Example - Problem 2: A polynomial function p(x) with real coefficients and of degree 5 has the zeros: -1, 2(with multiplicity 2) , 0 and 1. p(3) = -12.
Find p(x).

Solution to Problem 2:

p(x) can be written as follows

p(x) = ax(x + 1)(x - 2)^{2}(x - 1) , a is any real constant not equal to zero.

p(3) = -12 gives the following equation in a.

a(3)(3 + 1)(3 - 2)^{2}(3 - 1) = -12

Solve the above equation for a to obtain

a = -1/2

p(x) is given by

p(x) = -0.5x(x + 1)(x - 2)^{2}(x - 1)

The graph of p(x) is shown below.

Check the intercepts and the point (3 , -12) on the graph of p(x) found above.