Graphs of Third Degree Polynomials

The graphs of several third degree polynomials are shown along with questions and answers at the bottom of the page.

  1. Polynomial of a second degree polynomial: cuts the x axis at one point.

    Question 1: Why does the graph cut the x axis at one point only?

    Graph of a third degree polynomial, through the origin
    Figure 1: Graph of a third degree polynomial.



  2. Polynomial of a second degree polynomial: 3 x intercepts.

    Question 2: If the graph cuts the x axis at x = -2, what are the coordinates of the two other x intercpets?

    Graph of a third degree polynomial with 3 x intercepts.
    Figure 2: Graph of a third degree polynomial



  3. Polynomial of a third degree polynomial: 3 x intercepts and parameter a to determine.

    Question 3: The graph below cuts the x axis at x = 1 and has a y intercpet at y = 1. What are the coordinates of the two other x intercpets?

    Graph of a third degree polynomial with 3 x intercepts and one parameter to determine.
    Figure 3: Graph of a third degree polynomial



  4. Polynomial of a third degree polynomial: one x intercepts.

    Question 4: The graph below cuts the x axis at x = -1. Why does the graph of this polynomial have one x intercept only?

    Graph of a third degree polynomial with one x intercept.
    Figure 4: Graph of a third degree polynomial, one intercpet.





Answers to Above Questions
  1. Since x = 0 is a repeated zero or zero of multiplicity 3, then the the graph cuts the x axis at one point.

  2. An x intercept at x = -2 means that Since x + 2 is a factor of the given polynomial. Hence the given polynomial can be written as: f(x) = (x + 2)(x2 + 3x + 1). Find the other zero, which give the two other x intercpets, by solving the equation x2 + 3x + 1 = 0. The solutions are: x = -3/2 + SQRT(5) / 2 and x = -3/2 - SQRT(5) / 2.

  3. Use the y intercept to find a = 1 and then proceed in the same way as was done in question 2 above to find the other 2 x intercepts: 3/2 - SQRT(5) / 2 and 3/2 + SQRT(5) / 2

  4. Factor f as follows: f(x) = (x + 1)(x2 + x + 1). In solving the equation x2 + x + 1 = 0, the zeros are complex numbers and therefore do not show as x intercepts.


More references and links to polynomial functions.
Polynomial Functions


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Updated: 2 April 2013

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