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

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?

Figure 1: Graph of a third degree polynomial.

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?

Figure 2: Graph of a third degree polynomial

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?

Figure 3: Graph of a third degree polynomial

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?

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

Answers to Above Questions

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

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)(x^{2} + 3x + 1). Find the other zero, which give the two other x intercpets, by solving the equation x^{2} + 3x + 1 = 0. The solutions are: x = -3/2 + SQRT(5) / 2 and x = -3/2 - SQRT(5) / 2.

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

Factor f as follows: f(x) = (x + 1)(x^{2} + x + 1). In solving the equation x^{2} + x + 1 = 0, the zeros are complex numbers and therefore do not show as x intercepts.