Roots of Polynomials

Definitions:

• Polynomials are expressions involving x raised to a whole number power (exponent).

Some examples are:

• Roots of a polynomial are values of x which make the polynomial equal zero. For the examples of polynomials above, that means solving the following equations:

  
  
  

• Zeros of a polynomial: This is another term for “roots.”

1. If the roots are real, they are the x-intercepts on the graph of the polynomial.
2. If no roots are real (all roots are complex numbers of the form a + bi) then the polynomial graph does not intersect or cross the x-axis.
3. The roots relate directly to the factors. For example, if 3, 4 and 7 are the roots, then the factors of the polynomial are (x + 3), (x - 4), and (x - 7).

If the roots are not easy to find, locate the roots by using the TI-83 calculator under and the “2: zero” function. (LINK to CALCULATOR II lesson.)

Let's Practice: We will now examine five polynomial functions and their roots. In each case, the accompanying graph is shown under the discussion.

1. is a parabola LINK and its graph opens upward from the vertex . Note that since the graph does not cross the x-axis the solutions to   are not real numbers.

   

2.   is a parabola LINK and its graph opens downward from the vertex (1, 3). This graph was created by using the WINDOW (-5, 5) X (-8, 8). There are two x-intercepts and therefore two solutions to the equation . Solutions can be found using and the “2: zero” function. The x-intercepts are -0.22 and 2.22.

   

3. . This polynomial factors as The factors (x + 2) and (x- 3) make it clear that x = -2 and x = 3 are the roots or zeros of the parabola.

   

4. . This polynomial factors, but the factoring is not obvious. By examining the graph, we can see that the roots or zeros appear to be integers. That is a clue that this polynomial factors nicely. Using  and the "2: zero" function, we can verify that these roots are: -1, 2, and 4. This means that