Remainder Theorem

Finding the Remainder by Evaluating the Polynomial

The remainder of a polynomial division can often be found without completing long division or a Horner scheme. The Remainder Theorem connects that remainder directly to one value of the polynomial.

Statement of the Remainder Theorem

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

To find the remainder when $P(x)$ is divided by $(x-c)$ , evaluate $P(x)$ at $x=c$ .

For $P(x) = 2x^2 - 3x + 4$ divided by $x - 2$ , the remainder is $P(2) = 6$ , so you can answer the question with one substitution.

Why the Remainder Theorem Holds

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, 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 theorem determines the remainder directly and also explains why polynomial evaluation appears in division problems. Substituting the value that makes the divisor zero turns the division into a single evaluation, which is usually the faster route.

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

Using the Horner 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 $P(-4)$ .

Calculate $P(-4)$ :

\begin{aligned} P(-4) &= 2(-4)^5 + 5(-4)^4 \\ &\quad{}- 10(-4)^3 + 9(-4)^2 - 10 \end{aligned}

Both methods give the remainder $6$ . The Remainder Theorem reaches it without completing the full division scheme.

Evaluating $P(c)$ therefore gives exactly the remainder after 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. The divisor $x - 7$ corresponds to the value $x = 7$ , so one substitution is enough. Evaluate the polynomial at that value and report the resulting number as the remainder.

Worked Solution

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

\begin{aligned} P(7) &= 3(7)^5 - 20(7)^4 \\ &\quad{}- 6(7)^3 - 48(7) - 8 \end{aligned}

So, the remainder is $-1$ .