Graphing Polynomials

Graph polynomials ; a step by step tutorial with examples and detailed solutions. Factoring, zeros and their multiplicities, intercepts and other properties 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:
    (- ∞ , -2) , (- 2 , 0) , (0 , 1) and (1 , +∞)

    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 variable 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 , + ∞)


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.