Use the remainder theorem

For a polynomial $f(x)$ , division by the linear polynomial $x-a$ leaves the constant remainder $f(a)$ ; thus evaluating a polynomial at $a$ is equivalent to finding its remainder upon division by $x-a$ . This understanding supports efficient evaluation, interpretation of polynomial division, and the factor theorem: $x-a$ is a factor exactly when $f(a)=0$ . The scope is limited to linear divisors and standard real- or integer-coefficient polynomial examples, not higher-degree divisors or abstract coefficient systems.

Detailed Explanation: Use the remainder theorem

The remainder theorem says:

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

So, instead of performing long division, substitute $a$ into the polynomial.

Example

Find the remainder when

f(x)=2x^3-3x^2+4x-5

is divided by $x-2$ .

Step 1: Identify $a$

The divisor is $x-2$ , which has the form $x-a$ . Therefore,

a=2

Step 2: Evaluate $f(2)$

Substitute $2$ for $x$ :

f(2)=2(2)^3-3(2)^2+4(2)-5

Simplify:

f(2)=2(8)-3(4)+8-5

f(2)=16-12+8-5=7

Step 3: State the remainder

The remainder is

\boxed{7}

Therefore, dividing $2x^3-3x^2+4x-5$ by $x-2$ leaves a remainder of $7$ .

If $f(a)=0$ , then the remainder is zero, which means $x-a$ is a factor of $f(x)$ .

Learn by doing: Use the remainder theorem

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Factor Theorem - Find the Remainder Given a Binomial Divisor

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