Rewrite Rational Expressions

Lesson 6 of 7: Arithmetic with Polynomials

In this lesson:

• Identify proper vs. improper rational expressions
• Perform polynomial long division step by step
• Interpret zero remainder in terms of factoring

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Rewrite in the form by polynomial long division
2. Identify proper vs. improper rational expressions and explain when division is needed
3. Perform polynomial long division, correctly handling missing degree terms
4. Verify:
5. Recognize zero remainder as exact divisibility — a factoring consequence

Recall: Integer Long Division Steps

• Divide: leading term of dividend ÷ divisor → next quotient digit
• Multiply: quotient digit × divisor
• Subtract: remove that product from the dividend
• Repeat until remainder < divisor — write as quotient + remainder/divisor

What is in "mixed number" form?

Integer Analogy: The Division Algorithm

: we write , so

Same idea for polynomials:

Division reveals the "whole number part" and the "fractional remainder."

From the Remainder Theorem to Long Division

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: .

Proper vs. Improper Rational Expressions

Just as is proper (numerator < denominator) and is improper:

• Proper: — already in lowest form, no division needed
• Improper: — division required to extract the polynomial part

Rule: check degrees first. Divide only when the expression is improper.

Why You Need the Polynomial Part

An improper rational expression has a "polynomial part" hiding inside.

For graphing: the polynomial part determines long-run behavior (oblique asymptote).

Check-In: Classify These Rational Expressions

Classify each as proper or improper:

For each: compare degrees. No division yet — just classify.

Long Division Algorithm: Four Steps

1. Divide: leading term ÷ leading term → next quotient term
2. Multiply: entire divisor that term
3. Subtract: from current dividend
4. Stop: when

Worked Example: First Two Steps

Divide by :

• Divide:
• Multiply:
• Subtract: — degree 1, keep going

Worked Example: Complete and Verify

Divide: ; multiply: ; subtract: — stop (degree 0 < 1)

Verify: ✓

Probe: Verify the Division Result

A student claims:

Verify by computing .

Does it equal ?

Missing Terms: Write Placeholder Zeros

Divide by — missing term:

Write:

Complete division: quotient , remainder .

Placeholder Zeros: The Bookkeeping Rule

Step zero: write dividend with all missing terms as .

• →
• →

Higher-Degree Divisors: Same Algorithm, Wider Remainder

Stop when .

Worked: Placeholder Zeros + Zero Remainder

Divide by :

Write:

Result: — remainder !

Worked: Dividing by a Quadratic Expression

Dividend: . Three cycles → quotient .

Remainder: — degree 1 < 2, stop.

Zero Remainder Equals Exact Divisibility

Zero remainder = Factor Theorem for any degree divisor.

Worked: Simplify Using Zero Remainder

Dividend: — divide by .

Result: , remainder — is an exact factor.

Probe: Find the Placeholder Error

A student divides :

Setup: — no placeholder zeros
Divide: ; multiply:
Subtract: ← alignment error

What was skipped? Rework correctly.

Check-In: Full Division with Verification

Divide by :

1. Fill in the placeholder zero
2. Apply the algorithm to completion
3. Write as
4. Verify:

Three Division Errors to Avoid

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

Summary: Polynomial Long Division Algorithm

where

• When to use: — improper rational expression
• Algorithm: Divide → Multiply → Subtract → Repeat
• Stop: when
• Zero remainder: is a factor of ; simplify to

Next: Operations on Rational Expressions

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.