Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55

CAL2.37 Partial Fractions with Repeated Linear Factors Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.37 Partial Fractions with Repeated Linear Factors. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.37 Partial Fractions with Repeated Linear Factors - Select the correct terms needed for repeated linear factors in a partial-fraction decomposition. Standards: AP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Calculus II

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related CAL2.37 Partial Fractions with Repeated Linear Factors skill assessments:

Problem 1Difficulty: Mastery

For denominator (x - 2)³(x + 1)², a student writes A/(x - 2) + C/(x - 2)³ + D/(x + 1) + E/(x + 1)². Which single term is missing from the standard partial-fraction template?

AB/(x - 2)

BB/(x - 2)³

CB/(x - 2)⁴

DB/(x - 2)²

Solution & Explanation:The factor (x - 2) has multiplicity 3, so the template must include every power through (x - 2)³. The omitted term is B/(x - 2)²; skipping a middle or lower power makes the setup incomplete.

Problem 2Difficulty: Mastery

For distinct real numbers a and b, which symbolic template correctly represents the repeated-linear-factor part of a proper rational function with denominator (x - a)^m(x - b)ⁿ?

AA/(x - a)^m + B/(x - b)ⁿ

BA_1/(x - a) + ... + A_m/(x - a)^m + B/(x - b)ⁿ

CA_1/(x - a) + ... + A_m/(x - a)^m + B_1/(x - b) + ... + B_n/(x - b)ⁿ

DA/(x - a)^m + B_1/(x - b) + ... + B_n/(x - b)ⁿ

Solution & Explanation:Each repeated linear factor contributes a separate constant over every power from 1 to its multiplicity. The template therefore needs m terms for (x - a) and n terms for (x - b), not just the highest powers.

Problem 3Difficulty: Mastery

For denominator [(x - 1)²(x + 1)³]² = (x - 1)⁴(x + 1)⁶, what is the correct number of partial-fraction terms with constant numerators?

A9 constants

B10 constants, one for each required denominator power

C11 constants

D2 constants

Solution & Explanation:In standard partial fractions, the repeated factor (x - 1) with multiplicity 4 contributes powers (x - 1), (x - 1)², (x - 1)³, (x - 1)⁴; the repeated factor (x + 1) with multiplicity 6 contributes powers (x + 1), (x + 1)², (x + 1)³, (x + 1)⁴, (x + 1)⁵, (x + 1)⁶. Do not skip lower powers of a repeated linear factor. Therefore the template has 10 constants total.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Calculus II Math students. It aligns with educational standard AP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55 (Select the correct terms needed for repeated linear factors in a partial-fraction decomposition. Standards: AP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55).