In this lesson:
By the end of this lesson, you should be able to:
What is in "mixed number" form?
: we write , so
Same idea for polynomials:
Division reveals the "whole number part" and the "fractional remainder."
APR.B.2 (Remainder Theorem): gives the remainder when is divided by .
That worked for linear divisors only.
Today: polynomial long division — works for ANY divisor degree.
The quotient-and-remainder structure is the same: .
Just as is proper (numerator < denominator) and is improper:
Rule: check degrees first. Divide only when the expression is improper.
An improper rational expression has a "polynomial part" hiding inside.
For graphing: the polynomial part determines long-run behavior (oblique asymptote).
Classify each as proper or improper:
For each: compare degrees. No division yet — just classify.
Divide by :
Divide: ; multiply: ; subtract: — stop (degree 0 < 1)
Verify: ✓
A student claims:
Verify by computing .
Does it equal ?
Divide by — missing term:
Write:
Complete division: quotient , remainder .
Step zero: write dividend with all missing terms as .
Stop when .
Result: — remainder !
Before dividing by :
Setup first, prediction second — don't divide yet.
Dividend: . Three cycles → quotient .
Remainder: — degree 1 < 2, stop.
Zero remainder = Factor Theorem for any degree divisor.
Dividend: — divide by .
Result: , remainder — is an exact factor.
A student divides :
Setup: — no placeholder zeros Divide: ; multiply: Subtract: ← alignment error
What was skipped? Rework correctly.
Missing zeros: always write — no gaps allowed
Stopping too early: remainder degree must be strictly less than divisor degree
Dropping the remainder: answer is — not just
where
APR.D.7: adding, subtracting, and multiplying rational expressions.
To add : find a common denominator and combine.
The form from today makes simplification and asymptote analysis direct.
Click to begin the narrated lesson
Rewrite rational expressions