Question

The Remainder Theorem can be used as a shortcut to find the remainder when a function is divided by a binomial.

Answer the following questions in complete sentences.

Why is the Remainder Theorem used to determine whether a linear binomial is a factor of a polynomial?

How can you determine whether a linear binomial is a factor of a polynomial without using the Remainder Theorem?

Create your own division problem and show how to use the Remainder Theorem to determine whether a binomial is a factor of a polynomial function.

Answer 1

The Remainder Theorem is used to determine whether a linear binomial is a factor of a polynomial because it helps us factorize the polynomial more easily.

How to illustrate the theorem?

The remainder theorem states that if P(x) is a polynomial and x - a is a linear factor, the remainder when P(x) is divided by x - a is P(a). When P(a) = 0, then x - a is a factor of P(x).

Here, the remainder theorem is used to determine whether a linear binomial is a factor of a polynomial because it helps us factorize the polynomial more easily.

When the polynomial is divided by the linear factor, we obtain a polynomial of a lesser degree which can then be further factorized to obtain all the factors of our initial polynomial.

A linear binomial is the factor of a polynomial if the polynomial value is 0 at the zeros of the linear binomial

Let's assume a polynomial function is

P(x) = (x - 3)(x + 1)(x -2)

And a linear binomial is:

x - 3 = 0

We start by calculating the value of x in x - 3 = 0

l

x = 3

Next, we substitute x = 3 in P(x) = (x - 3)(x + 1)(x -2)

P(3) = (3 - 3)(3 + 1)(3 -2)

Evaluate

P(3) = 0

Since P(3) = 0, then the linear binomial x - 3 is a factor of P(x) = (x - 3)(x + 1)(x -2)

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