# Polynomial and Factored Form

Activity 4: Factor and Remainder Theorem

Content

In the last activity, you practiced the sketching of a polynomial graph, if you were given the Factored Form of the function statement. In this activity, you will learn a process for developing the Factored Form of a polynomial function, if given the General Form of the function.

Review

A polynomial function is a function whose equation can be expressed in the form of: f(x) = anxn + an-1xn-1 + an-2xn-2 + ...a2x2 + a1x + a0 where an, an-1, ...a1, a0 are real numbers and n is a natural number. The coefficient, which is attached to the highest degree of “x”, is called the leading coefficient.

A polynomial function expressed in the form f(x)= anxn + an-1xn-1 + an-2xn-2 + ...a2x2 + a1x + a0 is said to be in General Form.

Suppose you are given f(x) = anxn + an-1xn-1 + an-2xn-2 + ... a2x2 + a1x + a0 (this is a General Form of a polynomial function).

Your task is to sketch a graph of this function. Since you know how to sketch a graph of a polynomial function,if the function is in Factored Form, it would be handy for you to know how to move easily from General Form to Factored Form.

Tools

Before we continue with this task, you need some tools to help you in the process.

Dividing Polynomials.

Work through the following animation, to introduce you to the idea of dividing polynomials. You will learn two ways to do this! One is called Long Division, and the second is called Synthetic Division.

Watch to the clip titled Dividing Polynomials... but please be aware, that, depending on your Internet connection speed, the clip may take a few minutes to download. You can always continue reading the remainder of this page while you wait.

Finding the first factor.

Observation:

In the animation you just viewed, the examples illustrated how to divide a function, f(x), by a factor of the form (jx - k). Each of the solutions ended with a Division Statement. The Division Statement was always of the form: f(x) = Quotient . Divisor + Remainder

or

f(x) = q(x) . (x - k/j) + r

Suppose we were to evaluate f(k/j). Use the Division Statement form of the function to make it easier:

f(k/j) = q(k/j) . (k/j - k/j) + r

Because the value of (k/j - k/j) = 0, it is easy to see that, regardless of the value of q(k/j), we get this result:

f(k/j) = r

This is such a valuable result, that it has been given a name! This is called the Remainder Theorem.

The Remainder Theorem

When a polynomial function, f(x) is divided by (jx - k), the remainder, r, is f(k/j).

This simplifies if j = 1 i.e.: we are dividing by (x - k).

When a polynomial function, f(x) is divided by (x - k), the remainder is f(k).

Take this idea one step further. We are looking for FACTORS of a polynomial. A factor of a polynomial divides evenly into the polynomial, with a remainder of zero.

So, in the special case of the Remainder Theorem - the case when the remainder is zero - we know the divisor is a factor of the dividend.

This is such an important observation that it has been given a name: The Factor Theorem.

The Factor Theorem

A polynomial function, f(x), has a factor of (x - k) if and only if f(k) = 0.

So, if f(k) = 0, then (x-k) is a factor of f(x).

Similarly

(jx-k) is a factor of f(x) if and only if f(k/j) = 0.

Now we can return to our task: changing the General Form of a function statement into the corresponding Factored Form of the function statement.

We will go through the process with an example, and then stop to consider a general set of steps we can use for any such situation.

Example 1:

Given: f(x) = x³ - 4x² + x + 6

Task: Find the fully Factored Form of the function statement (often in preparation for another task - for example, a graph sketch - though we will concentrate on the factoring for now.)

Solution:

We need a factor to start with. We know the Factor Theorem suggests we would like to...

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