Polynomial Functions, Zeros, Factors and Intercepts (1)

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

1. s is a zero for the polynomial function p(x).
   
   
2. s is a solution to the equation p(x) = 0
   
   
3. x - s is a factor of p(x).
   
   
4. The point (s , 0) is an x intercept of the graph of p(x).
   
   

Example: let p(x) = x³ - 2·x² + 2·x - 4

1. p(2) = 2³ - 2*2² + 2*2 - 4
   
   = 8 - 8 + 4 - 4
   = 0
   2 is a zero of p(x).
   
   
2. x = 2 is a solution of p(x) = 0
   
   
3. p(x) can be written in factored form as
   p(x) =(x - 2)·(x² + 2)
   
   
4. 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:

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.
   
   
2. 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
   
   
3. Solve the above equation for a to obtain
   a = 1
   
   
4. p(x) is given by
   p(x) = (x + 1)²(x - 2)
   
   

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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:

1. 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.
   
   
2. p(3) = -12 gives the following equation in a.
   a(3)(3 + 1)(3 - 2)²(3 - 1) = -12
   
   
3. Solve the above equation for a to obtain
   a = -1/2
   
   
4. p(x) is given by
   p(x) = -0.5x(x + 1)(x - 2)²(x - 1)
   
   
   
5. The graph of p(x) is shown below.
   Check the intercepts and the point (3 , -12) on the graph of p(x) found above.
   
   

More references and links to polynomial functions.

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Factor Polynomials.

Polynomial Functions.

Multiplicity of Zeros and Graphs Polynomials.

Find Zeros of Polynomial Functions - Problems

Graphs of Polynomial Functions - Self Test.