Polynomial Division and the Remainder Theorem

What is left when a factor does not fit.

12 problems · about 22 minutes · AP Precalculus 1.9

NameDate

Figure — use these to answer the problems

Each real root is a crossing. Finding one lets you divide it out and shrink the polynomial.

y = 1x³ − 4x + 0

b-4

Figure 1

Straightforward practice. Get the method working first.

5 problems

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

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
f(x) = x² − 5x + 6. Remainder on division by (x − 3)?
Answer
f(x) = 2x + 7. Remainder on division by (x + 1)?
Answer

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

4 problems

9
Match each statement with what it tells you.
Draw a line from each item on the left to its match on the right.
- The remainder equals f(a)
- f(a) is zero
- The numerator degree exceeds the denominator
- The Remainder Theorem
- (x minus a) is a factor
- Division reveals a slant or polynomial asymptote
f(x) = x³ + 1. Remainder on division by (x + 1)?
Answer
The Remainder
f(x) = x³ − 2x + 5 divided by (x − 2). What is the remainder?
Answer
The Factor
f(3) = 0. Is (x − 3) a factor? 1 yes, 0 no.
Answer