Cogito
Algebra 2 · Chapter 3 · Lesson 2
Dividing Polynomials
Long division, and the shortcut that replaces it.
11 problems · about 23 minutes · A-APR.B.2, A-APR.D.6
What this lesson teaches
The student divides polynomials and uses the Remainder Theorem.
- Polynomial long division mirrors the numeric kind.
- The remainder on dividing by (x − a) is simply f(a).
- A zero remainder means (x − a) is a factor and a is a root.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(x) = x² + 2, divided by (x − 3). Remainder?
Answer 11
Why 11.
f(x) = x² − 5, divided by (x − 2). Remainder?
Answer -1
Why f(2) = 4 − 5.
f(x) = x³, divided by (x − 1). Remainder?
Answer 1
Why f(1).
f(x) = x² − 9, divided by (x − 3). Remainder?
Answer 0
Why f(3) = 0.
Build It Up
The same ideas with more to keep track of.
3 problemsIs (x − 3) therefore a factor of x² − 9? 1 for yes, 0 for no.
Answer 1
Why Zero remainder.
f(x) = 2x + 6, divided by (x + 3). Remainder?
Answer 0
Why f(−3).
Degree 5 divided by degree 2. What is the degree of the quotient?
Answer 3
Why Subtract the degrees.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsf(x) = x² + x, divided by (x − 2). Remainder?
Answer 6
Why 4 + 2.
Select every statement that follows from f(a) = 0.
Answer (x − a) is a factor.; a is a root.
Why A root and a factor are the same fact stated two ways.
The Remainder: f(x) = x² + 1 divided by (x − 3). What is the remainder?
Answer 10
Why 10.
The Factor: f(4) = 0. Is (x − 4) a factor? 1 for yes, 0 for no.
Answer 1
Why Yes.