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

P(c) = (0) \cdot H(c) + S

P(c) = 0 + S

P(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

P(-4) = 2(-1024) + 5(256) - 10(-64) + 9(16) - 10

P(-4) = -2048 + 1280 + 640 + 144 - 10

P(-4) = -2048 + 1920 + 144 - 10

P(-4) = -128 + 144 - 10

P(-4) = 16 - 10

P(-4) = 6

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

P(7) = 3(16807) - 20(2401) - 6(343) - 336 - 8

P(7) = 50421 - 48020 - 2058 - 336 - 8

P(7) = 2401 - 2058 - 336 - 8

P(7) = 343 - 336 - 8

P(7) = 7 - 8

P(7) = -1

So, the remainder is $-1$ .