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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.

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 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 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

Examples
Example Let . Find the roots of P(x). What are the factors of P(x)?
What is your answer?
 
Example Let  . Find the roots of P(x). What are the factors of P(x)?
What is your answer?
 
Example
Let Find the roots of P(x). What are the factors of P(x)?
What is your answer?
 
Example Let . Find the roots of P(x). What are the factors of P(x)?
What is your answer?
 
Example Let . Find the roots of P(x). What are the factors of P(x)?
What is your answer?
 



M Ransom

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