LEARN · EXPLAIN · REVISE

Decompose the rational expression with its repeated denominator factor.

Read the idea, work independently, then explain what changed.

2027 JM01 考試大綱 · 4. Polynomial and rational fractions · PDF 2 / printed page 2

TOPIC 01

Polynomial division and partial fractions

Divide polynomials and choose decomposition numerators matched to each denominator factor.

Divide polynomials and choose decomposition numerators matched to each denominator factor.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Worked example

(x+3)/x^2

Record all original excluded roots. Divide improper fractions first; a repeated factor needs every power.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use this intermediate relation.

A/x+B/x^2

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$Ax+B=x+3\Rightarrow A=1,\ B=3$
Both x and x² terms appear; omitting one cannot match the numerator.

The requested relation or conclusion is shown below.

1/x+3/x^2

Checks and common pitfalls: Both x and x² terms appear; omitting one cannot match the numerator.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Divide polynomials and choose decomposition numerators matched to each denominator factor.
Which condition is essential in polynomial division and partial fractions?
Can a canceled factor still create a missing point in the graph?

Model or definition: Divide polynomials and choose decomposition numerators matched to each denominator factor.
$R(x)=Q(x)+A/(x-a)+B/(x-b)$
Conditions: Record all original excluded roots. Divide improper fractions first; a repeated factor needs every power.