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


  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 - 2x2 + 2x - 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.

    Graph of p(x), example above.




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).
Graph of p(x), problem 1.

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)





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.

    Graph of p(x), problem 2.
More references and links to polynomial functions.
Factor Polynomials.

Polynomial Functions.

Multiplicity of Zeros and Graphs Polynomials.

Find Zeros of Polynomial Functions - Problems

Graphs of Polynomial Functions - Self Test.

Step by Step Solver to Find a Polynomial Given its Zeros and a Point.

Step by Step Solver to Factor a Cubic Polynomial Given one of its Zeros.

Find a Cubic Polynomial Passing Through Four Points.

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