91Ó°ÊÓ

A polynomial equation expressed as a product of its factors is in its factored form. If you know the roots of a polynomial, you can easily write the polynomial in this form.

For instance, let's say we have a third-degree polynomial with roots ewline -2, 1, and 5. Each root translates to a factor in the form of ewline ewline (x-root).

Given the roots -2, 1, and 5:

• -2 becomes (x + 2)
• 1 becomes (x - 1)
• 5 becomes (x - 5)

Hence, the factored form of the polynomial is:

ewline

ewline ewline (x + 2)(x - 1)(x - 5).
This form makes it easy to identify the roots directly by setting each factor to zero and solving for x.

Expanding factors involves multiplying them out to yield the standard polynomial form. This process transitions the polynomial from its factored form to an expanded form, which is typically more familiar.

Let's expand the factors from the previous section step by step.

First, we start with the first two factors: ewline ewline (x + 2)(x - 1).
Multiply them together:
ewline (x + 2)(x - 1) = x^2 - x + 2x - 2 = x^2 + x - 2.

Now, we incorporate the third factor (x - 5): ewline ewline (x^2 + x - 2)(x - 5).

Distribute all terms in the first polynomial by each term in the second polynomial: ewline ewline x^2(x - 5) + x(x - 5) - 2(x - 5) ewline ewline = x^3 - 5x^2 + x^2 - 5x - 2x + 10.

Combine like terms to get the expanded form:
ewline ewline x^3 - 4x^2 - 7x + 10.

The roots of a polynomial are the values of x that make the polynomial equal to zero. For a third-degree polynomial, you can expect up to three roots.

Given our polynomial in its expanded form: ewline ewline x^3 - 4x^2 - 7x + 10 = 0,
we know that these roots were obtained from the original factored form: ewline ewline (x + 2)(x - 1)(x - 5) = 0.
Setting each factor to zero:

• x + 2 = 0 → x = -2
• x - 1 = 0 → x = 1
• x - 5 = 0 → x = 5

Therefore, the roots of the polynomial are <-2, 1, and 5>. Knowing these roots and the process of factoring and expanding helps in understanding the behavior of the polynomial function as well as solving it.