Remainder Theorem

Remainder Theorem: how to use it

The Remainder Theorem connects division of a polynomial by a linear expression to simple substitution. If a polynomial (f(x)) is divided by ((x-a)), the remainder is (f(a)). So instead of doing full polynomial long division, you can often find the remainder by evaluating the polynomial at the correct value.

Step-by-step method

1. Identify the divisor.
   • If the divisor is (x-a), use (a).
   • If the divisor is (x+a), rewrite it as (x-(-a)), so use (-a).
2. Substitute the value into the polynomial.
3. Simplify carefully.
   • Combine like terms.
   • Watch signs and exponents.
4. State the result as the remainder.
   • If the value is 0, the polynomial is divisible by the divisor.

Quick check

Make sure the number you plug in matches the factor exactly after rewriting it in the form (x-a). Your final answer should be the evaluated value, fully simplified.

Common mistake to avoid

Do not set the divisor equal to zero and solve for (x) in the middle of the process unless you are identifying the needed substitution value. The remainder comes from evaluating the polynomial, not from dividing the coefficients by hand unless the problem specifically asks for it.