Remainder Theorem: Polynomial - Mathematics (Grade 11)

Understanding the Remainder Theorem

Have you ever wondered if there's a quick way to find the remainder of a polynomial division without performing long division or the lengthy Horner's method? The answer lies in the Remainder Theorem!

The Remainder Theorem provides an interesting relationship between the remainder of a polynomial division and the value of the polynomial itself.

Statement of the Remainder Theorem

If a polynomial $P(x)$ is divided by $(x - c)$ , then the remainder is $S = P(c)$ .

This means that to find the remainder when $P(x)$ is divided by $(x - c)$ , we simply need to evaluate the polynomial $P(x)$ at $x = c$ .

Why Does the Remainder Theorem Work?

This theorem stems directly from the polynomial division algorithm we already know:

P(x) = (x - c) \cdot H(x) + S

Where:

$P(x)$ is the dividend polynomial.
$(x - c)$ is the divisor polynomial (degree $1$ ).
$H(x)$ is the quotient.
$S$ is the remainder (a constant, since the divisor is degree $1$ ).

Now, let's substitute $x = c$ into the division algorithm equation:

P(c) = (c - c) \cdot H(c) + S

This proves that the value of the polynomial $P(x)$ at $x = c$ is equal to the remainder $S$ when $P(x)$ is divided by $(x - c)$ .

Calculating with the Remainder Theorem

The Remainder Theorem is very useful for quickly determining the remainder of a division or for evaluating a polynomial at a specific point.

Find the remainder if $P(x) = 2x^5 + 5x^4 - 10x^3 + 9x^2 - 10$ is divided by $x + 4$ .

Using Horner's Method

The divisor is $x + 4$ , or $x - (-4)$ , so $c = -4$ .

Coefficients of $P(x)$ (completing the x term): $2, 5, -10, 9, 0, -10$ .

\begin{array}{c|cccccc} -4 & 2 & 5 & -10 & 9 & 0 & -10 \\ & & -8 & 12 & -8 & -4 & 16 \\ \hline & 2 & -3 & 2 & 1 & -4 & \boxed{6} \\ \end{array}

From Horner's method, we get:

Quotient: $H(x) = 2x^4 - 3x^3 + 2x^2 + x - 4$
Remainder: $S = \boxed{6}$

Using the Remainder Theorem

According to the Remainder Theorem, the remainder when $P(x)$ is divided by $x - (-4)$ is $P(-4)$ .

Let's calculate $P(-4)$ :

P(-4) = 2(-4)^5 + 5(-4)^4 - 10(-4)^3 + 9(-4)^2 - 10

The result is the same! Using the Remainder Theorem, we found the remainder is $6$ , just like with Horner's method, but without performing the full division process.

This shows that evaluating $P(c)$ is another way to find the remainder of division by $(x-c)$ .

Exercise

If $P(x) = 3x^5 - 20x^4 - 6x^3 - 48x - 8$ is divided by $x - 7$ , determine the remainder using the Remainder Theorem.

Answer Key

According to the Remainder Theorem, the remainder when $P(x)$ is divided by $x - 7$ is $P(7)$ .

P(7) = 3(7)^5 - 20(7)^4 - 6(7)^3 - 48(7) - 8

So, the remainder is $-1$ .