Polynomial Division and the Remainder Theorem

What is left when a factor does not fit.

12 problems · about 22 minutes · AP Precalculus 1.9

What this lesson teaches

The student divides polynomials and applies the Remainder and Factor Theorems.

Dividing by (x − a) leaves a remainder of f(a).
A zero remainder means (x − a) is a factor.
Division rewrites a top-heavy rational function as a polynomial plus a proper fraction.

Straightforward practice. Get the method working first.

5 problems

f(x) = x² + 2x. Remainder on division by (x − 3)?
Answer 15
Why 15.
Why is the Factor Theorem a special case of the Remainder Theorem?
Answer A factor means a remainder of zero, and the remainder is f(a).
Why It is the remainder theorem with remainder zero.
f(x) = x² + 3x. Remainder on division by (x − 1)?
Answer 4
Why f(1) = 1 + 3.
f(x) = x³ − 8. Remainder on division by (x − 2)?
Answer 0
Why f(2) = 8 − 8.
That remainder means (x − 2) is a factor? 1 yes, 0 no.
Answer 1
Why A zero remainder.

The same ideas with more to keep track of.

3 problems

A degree-4 polynomial divided by a linear factor. Degree of the quotient?
Answer 3
Why One lower.
f(x) = x² − 5x + 6. Remainder on division by (x − 3)?
Answer 0
Why f(3) = 9 − 15 + 6.
f(x) = 2x + 7. Remainder on division by (x + 1)?
Answer 5
Why f(−1) = −2 + 7.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each statement with what it tells you.
Answer The remainder equals f(a) → The Remainder Theorem; f(a) is zero → (x minus a) is a factor; The numerator degree exceeds the denominator → Division reveals a slant or polynomial asymptote
Why One of these is the special case of the other.
f(x) = x³ + 1. Remainder on division by (x + 1)?
Answer 0
Why f(−1) = −1 + 1.
The Remainder: f(x) = x³ − 2x + 5 divided by (x − 2). What is the remainder?
Answer 9
Why 9.
The Factor: f(3) = 0. Is (x − 3) a factor? 1 yes, 0 no.
Answer 1
Why Yes.