Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Polynomial division and partial fractions

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

Revisit first: Function concept and representations

TOPIC 01

Polynomial division and partial fractions

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

What you will be able to explain

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

Model or definition

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

R(x)=Q(x)+A/(x−a)+B/(x−b)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.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Can a canceled factor still create a missing point in the graph?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

The difference is 2ab=4. Equality holds exactly when ab=0.(a+b)² = 9a²+b² = 5

The difference is 2ab=4. Equality holds exactly when ab=0.

Explain: Compare two admissible cases and explain their different results using the stated model.

Transfer: Compare equality as rational expressions with equality of functions on their original domains.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find the remainder when x²+t is divided by x−2.

t=2t=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.
R=f(2)R=f(2)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    R=22+2=6R=2^2+2=6

  3. The remainder theorem evaluates at the zero of the divisor.

The requested value is 6.

Checks and common pitfalls: The remainder theorem evaluates at the zero of the divisor.

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

02 / Standard#Worked example

Decompose the rational expression with its repeated denominator factor.

(x+3)/x2(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/x2A/x+B/x^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    Ax+B=x+3⇒A=1, B=3Ax+B=x+3\Rightarrow A=1,\ B=3

  3. Both x and x² terms appear; omitting one cannot match the numerator.

The requested relation or conclusion is shown below.

1/x+3/x21/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
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

03 / Transfer#Worked example

Explain the difference between f and g as functions.

f(x)=(x2−16)/(x−4),g(x)=x+4f(x)=(x^2-16)/(x-4),\quad g(x)=x+4
  • 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.
x2−t2=(x−t)(x+t)x^2-t^2=(x-t)(x+t)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f(x)=x+4(x≠4)f(x)=x+4\quad(x\ne4)

  3. The simplified expression agrees on the shared domain but does not fill the hole in f.

The requested relation or conclusion is shown below.

f(x)=g(x) (x≠4);f(4) undefinedf(x)=g(x)\ (x\ne4);\quad f(4)\text{ undefined}

Checks and common pitfalls: The simplified expression agrees on the shared domain but does not fill the hole in f.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

04 / Foundation#Your turn

Find the remainder when x²+t is divided by x−2.

t=5t=5
  • 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.
R=f(2)R=f(2)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    R=22+5=9R=2^2+5=9

  3. The remainder theorem evaluates at the zero of the divisor.

The requested value is 9.

Checks and common pitfalls: The remainder theorem evaluates at the zero of the divisor.

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

05 / Foundation#Your turn

Divide x²+t x+1 by x.

t=6t=6
  • 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
Divide each term by x.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    (x2+6x+1)/x=x+6+1/x(x^2+6x+1)/x=x+6+1/x

  3. The original domain excludes x=0.

The requested relation or conclusion is shown below.

x+6+1/xx+6+1/x

Checks and common pitfalls: The original domain excludes x=0.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

06 / Foundation#Your turn

In t/[x(x+1)]=A/x+B/(x+1), find A.

t=7t=7
  • 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.
t=A(x+1)+Bxt=A(x+1)+Bx
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    x=0⇒A=7x=0\Rightarrow A=7

  3. Apply the stated relation and retain its conditions.

    B=−7B=-7

  4. Compare polynomial identities after multiplying denominators; the rational identity still excludes 0 and −1.

The requested value is 7.

Checks and common pitfalls: Compare polynomial identities after multiplying denominators; the rational identity still excludes 0 and −1.

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

07 / Foundation#Your turn

In (x+t)/[(x−1)(x+1)]=A/(x−1)+B/(x+1), find A.

t=8,(x+8)/[(x−1)(x+1)]t=8,\quad (x+8)/[(x-1)(x+1)]
  • 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.
x+t=A(x+1)+B(x−1)x+t=A(x+1)+B(x-1)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    x=1⇒1+8=2Ax=1\Rightarrow 1+8=2A

  3. Apply the stated relation and retain its conditions.

    A=4.5A=4.5

  4. Evaluating the cleared polynomial identity at x=1 is permitted even though the original fraction excludes that input.

The requested value is 4.5.

Checks and common pitfalls: Evaluating the cleared polynomial identity at x=1 is permitted even though the original fraction excludes that input.

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

08 / Standard#Your turn

In (x+t)/[(x−1)(x+1)]=A/(x−1)+B/(x+1), find A.

t=9,(x+9)/[(x−1)(x+1)]t=9,\quad (x+9)/[(x-1)(x+1)]
  • 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.
x+t=A(x+1)+B(x−1)x+t=A(x+1)+B(x-1)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    x=1⇒1+9=2Ax=1\Rightarrow 1+9=2A

  3. Apply the stated relation and retain its conditions.

    A=5A=5

  4. Evaluating the cleared polynomial identity at x=1 is permitted even though the original fraction excludes that input.

The requested value is 5.

Checks and common pitfalls: Evaluating the cleared polynomial identity at x=1 is permitted even though the original fraction excludes that input.

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

09 / Standard#Your turn

Decompose the rational expression with its repeated denominator factor.

(x+10)/x2(x+10)/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/x2A/x+B/x^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    Ax+B=x+10⇒A=1, B=10Ax+B=x+10\Rightarrow A=1,\ B=10

  3. Both x and x² terms appear; omitting one cannot match the numerator.

The requested relation or conclusion is shown below.

1/x+10/x21/x+10/x^2

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

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

10 / Standard#Your turn

Find quotient and remainder, then express the improper fraction.

(x2+11)/(x−1)(x^2+11)/(x-1)
  • 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.
x2+t=(x−1)(x+1)+(t+1)x^2+t=(x-1)(x+1)+(t+1)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    (x−1)(x+1)=x2−1(x-1)(x+1)=x^2-1

  3. Apply the stated relation and retain its conditions.

    R=12R=12

  4. A polynomial part is needed before partial fractions of an improper fraction.

The requested relation or conclusion is shown below.

x+1+12/(x−1)x+1+12/(x-1)

Checks and common pitfalls: A polynomial part is needed before partial fractions of an improper fraction.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

11 / Standard#Your turn

Explain the difference between f and g as functions.

f(x)=(x2−144)/(x−12),g(x)=x+12f(x)=(x^2-144)/(x-12),\quad g(x)=x+12
  • 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.
x2−t2=(x−t)(x+t)x^2-t^2=(x-t)(x+t)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f(x)=x+12(x≠12)f(x)=x+12\quad(x\ne12)

  3. The simplified expression agrees on the shared domain but does not fill the hole in f.

The requested relation or conclusion is shown below.

f(x)=g(x) (x≠12);f(12) undefinedf(x)=g(x)\ (x\ne12);\quad f(12)\text{ undefined}

Checks and common pitfalls: The simplified expression agrees on the shared domain but does not fill the hole in f.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

12 / Transfer#Your turn

Find quotient and remainder, then express the improper fraction.

(x2+13)/(x−1)(x^2+13)/(x-1)
  • 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.
x2+t=(x−1)(x+1)+(t+1)x^2+t=(x-1)(x+1)+(t+1)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    (x−1)(x+1)=x2−1(x-1)(x+1)=x^2-1

  3. Apply the stated relation and retain its conditions.

    R=14R=14

  4. A polynomial part is needed before partial fractions of an improper fraction.

The requested relation or conclusion is shown below.

x+1+14/(x−1)x+1+14/(x-1)

Checks and common pitfalls: A polynomial part is needed before partial fractions of an improper fraction.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

13 / Transfer#Your turn

Explain the difference between f and g as functions.

f(x)=(x2−196)/(x−14),g(x)=x+14f(x)=(x^2-196)/(x-14),\quad g(x)=x+14
  • 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.
x2−t2=(x−t)(x+t)x^2-t^2=(x-t)(x+t)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f(x)=x+14(x≠14)f(x)=x+14\quad(x\ne14)

  3. The simplified expression agrees on the shared domain but does not fill the hole in f.

The requested relation or conclusion is shown below.

f(x)=g(x) (x≠14);f(14) undefinedf(x)=g(x)\ (x\ne14);\quad f(14)\text{ undefined}

Checks and common pitfalls: The simplified expression agrees on the shared domain but does not fill the hole in f.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

Focus on one question

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • 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?

    Board plan

    • Model or definition: Divide polynomials and choose decomposition numerators matched to each denominator factor.
      R(x)=Q(x)+A/(x−a)+B/(x−b)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.

    Anticipated thinking

    • Canceling a factor does not restore an excluded point to the original domain.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

    No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

    Curriculum and source notes ↗