Graphing Polynomials

Graph polynomials : a step by step tutorial with examples and detailed solutions. Factoring, zeros and intercepts are used to graph polynomials.


Example 1:

a) Factor polynomial P given by


P (x) = - x3 - x2 + 2x

b) Determine the multiplicity of each zero of P.

c) Determine the sign chart of P.

d) Graph polynomial P and label the x and y intercepts on the graph obtained.

Solution to Example 1

  • a) Factor P as follows

    P (x) = - x3 - x2 + 2x

    = - x (x2 + x - 2)

    = - x (x + 2)(x - 1)

    b) P has three zeros which are -2, 0 and 1 and are all of multiplicity one.

    c) The three zeros of P will split the number line into four intervals given by:

    (-infinity , -2) , (- 2 , 0) , (0 , 1) and (1 , +infinity)

    select one value of x within each interval and evaluate polynomial P for this value to determine the sign of P.

    intervals on number line


    P(-3) = 12 > 0 , P(-1) = - 2 < 0 , P(1/2) = 5/8 >0 , P(2) = - 8 < 0

    Using the signs of P with each interval, the sign chart is as follows:

    values of varibale x within intervals


    Graph of polynomial P in example 1


Example 2:

a) Factor polynomial P given by


P (x) = x4 - 2 x2 + 1

b) What is the multiplicity of each zero of P?

c) Determine the sign chart of P.

d) Graph polynomial P and label the x and y intercepts on the graph obtained.

e) What is the range of polynomial P?

Solution to Example 2

  • a) Factor P as follows

      P (x) = x4 - 2 x2 + 1

      = (x2 - 1)2


    = ((x - 1)(x + 1))2

    = (x - 1)2 (x + 1)2

  • b) Polynomial P has zeros at x = 1 and x = -1 and both has multiplicity 2.

    c) Polynomial P(x) is a perfect square and therefore positive or zero for all real values of x. P(x) is equal to zero at the two zeros -1 and 1 and positive everywhere else. The sign chart is as follows:

    sign chart of polynomial in example 2


    d) The x intercepts are at (-1,0) and (1,0) and the y intercept is at (0,1). The graph of P touches the x axis at x = -1 and x = 1 and opens up since P(x) is positive, cutting the y axis at (0,1). See graph below.

    Graph of polynomial example 2


    e) Using the graph of P above, the range of P is given by the interval

    [ 0 , + infinity)


Example 3:

a) Show that x = - 3 is a zero of polynomial P given by


P (x) = x4 + 5 x3 + 5 x2 - 5 x - 6

b) Show that (x - 1) is a factor of P.

c) Factor P and determine the multiplicity of each zero of P.

d) Determine the sign chart of P.

e) Graph polynomial P and label the x and y intercepts on the graph obtained.

Solution to Example 3

  • a) Calculate P(-3)

    P (-3) = (-3)4 + 5 (-3)3 + 5 (-3)2 - 5 (-3) - 6

    = 0
  • Hence -3 is a zero of P and x + 3 is a factor of P(x).

  • b) Since (x + 3) is a factor of P(x), the division of P(x) by (x + 3) must give a remainder equal to zero. Hence

    P(x) / (x + 3) = x3 + 2 x2 - x - 2

    We now divide x3 + 2 x2 - x - 2 by (x - 1)

    (x3 + 2 x2 - x - 2) / (x - 1) = x2 + 3x + 2 , remainder equal to zero which proves that (x - 1) is a factor of x3 + 2 x2 - x - 2 and therefore of P(x).

  • c) Using the above, P(x) may be written as follows

    P(x) = (x + 3)(x - 1)(x2 + 3x + 2)

    We now factor the quadratic term x2 + 3x + 2 included in P(x). Hence

    P(x) = (x + 3)(x - 1)(x + 1)(x + 2)

    P has zeros at x = -3, -2, -1 and 1 and are all of multiplicity one.

  • d) The sign chart is shown below

    Sign chart of polynomial in example 3


  • e) Using the information on the zeros and the sign chart, the graph of P is as shown below with x and y intercepts labeled.

    graph of polynomial in example 3


Example 4:

x = 1 is a zero of multiplicity 2 of polynomial P defined by

P (x) = x5 + x4 - 3 x3 - x2 + 2 x.

Construct a sign chart for P and graph it.

Solution to Example 4

  • If x = 1 is a zero of multiplicity 2, then (x - 1)2 is a factor of P(x) and a division of P(x) by (x - 1)2 yields a remainder equal to 0. Hence

    P (x) / (x - 1)2

    = (x5 + x4 - 3 x3 - x2 + 2 x) / (x - 1)2

    = x3 + 3 x 2 + 2 x

    P(x) is now factored as follows

    P(x) = (x - 1)2 (x3 + 3 x 2 + 2 x)

    = x (x - 1)2 (x2 + 3 x + 2 )

    = x (x - 1)2 (x + 1)(x + 2)

    P(x) has 4 zeros at x = -2, -1, 0 and 1 and the zero at x= 1 is of multiplicity 2.

    The sign chart is shown below

    Sign chart of polynomial in example 4


    Use the sign chart and the zeros of P to grpah P as shown below.

    graph of polynomial in example 4



More references and links to Graphing Functions.






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