Question #8044

What method is used to determine the zeros of a polynomial function of degree three or greater? Explain your answer and provide an example.

Expert's answer

Let's consider the following third degree polynomial function:

f ( x ) = x³ + 2x² - x - 2

Find f ( 1 ), f ( -1 ) and f ( -2 ) .

Solution

Now notice that the polynomial can be factored by grouping.

f ( x ) = x²(x + 2) - (x + 2) = (x² - 1)(x + 2) = (x - 1) (x + 1)(x - 2).

This example illustrates an important principle about polynomial functions: the number c is a zero of a polynomial function if and only if (x - c) is a factor of the polynomial.

So, in principle, the problem of finding the zeros of a polynomial function and the problem of factoring the polynomial are the same problem.

One of the main reasons we try to find the zeros of a polynomial function is so that we may factor the polynomial.

f ( x ) = x³ + 2x² - x - 2

Find f ( 1 ), f ( -1 ) and f ( -2 ) .

Solution

Now notice that the polynomial can be factored by grouping.

f ( x ) = x²(x + 2) - (x + 2) = (x² - 1)(x + 2) = (x - 1) (x + 1)(x - 2).

This example illustrates an important principle about polynomial functions: the number c is a zero of a polynomial function if and only if (x - c) is a factor of the polynomial.

So, in principle, the problem of finding the zeros of a polynomial function and the problem of factoring the polynomial are the same problem.

One of the main reasons we try to find the zeros of a polynomial function is so that we may factor the polynomial.

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