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

CAL2.39 Partial Fractions with Mixed 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 CAL2.39 Partial Fractions with Mixed Factors. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.39 Partial Fractions with Mixed Factors - Select the correct full decomposition for a rational function with mixed linear and quadratic factors. Standards tags: 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 CAL2.39 Partial Fractions with Mixed Factors skill assessments:

Problem 1Difficulty: Mastery

For the rational function (21x⁴ + 14x³ - 29x² - 46x + 1)/((4x - 1)(x + 1)²(x² + 2x + 2)), which partial fraction decomposition is correct over the real numbers?

A3/(4x - 1) + 2/(x + 1) + (-5)/(x + 1)² + (4x - 1)/(x² + 2x + 2)

B(-3)/(4x - 1) + 3/(x + 1) + (-5)/(x + 1)² + (4x - 1)/(x² + 2x + 2)

C(-3)/(4x - 1) + 2/(x + 1) + (-5)/(x + 1)² + (4x - 1)/(x² + 2x + 2)

D(-4)/(4x - 1) + 2/(x + 1) + (-5)/(x + 1)² + (4x - 1)/(x² + 2x + 2)

Solution & Explanation:Clear the common denominator (4x - 1)(x + 1)²(x² + 2x + 2). The decomposition (-3)/(4x - 1) + 2/(x + 1) + (-5)/(x + 1)² + (4x - 1)/(x² + 2x + 2) produces the numerator 21x⁴ + 14x³ - 29x² - 46x + 1, so it equals the given rational function. The irreducible quadratic factors need linear numerators, and any repeated factor needs a term for each power.

Problem 2Difficulty: Mastery

For the rational function (-x⁵ + 16x⁴ - 106x³ + 357x² - 645x + 554)/((x - 1)²(x² - 4x + 8)²), which partial fraction decomposition is correct over the real numbers?

A2/(x - 1) + 7/(x - 1)² + (x - 3)/(x² - 4x + 8) + (-5x + 2)/(x² - 4x + 8)²

B(-2)/(x - 1) + 8/(x - 1)² + (x - 3)/(x² - 4x + 8) + (-5x + 2)/(x² - 4x + 8)²

C(-2)/(x - 1) + 7/(x - 1)² + (x - 3)/(x² - 4x + 8) + (-5x + 2)/(x² - 4x + 8)²

D(-3)/(x - 1) + 7/(x - 1)² + (x - 3)/(x² - 4x + 8) + (-5x + 2)/(x² - 4x + 8)²

Solution & Explanation:Clear the common denominator (x - 1)²(x² - 4x + 8)². The decomposition (-2)/(x - 1) + 7/(x - 1)² + (x - 3)/(x² - 4x + 8) + (-5x + 2)/(x² - 4x + 8)² produces the numerator -x⁵ + 16x⁴ - 106x³ + 357x² - 645x + 554, so it equals the given rational function. The irreducible quadratic factors need linear numerators, and any repeated factor needs a term for each power.

Problem 3Difficulty: Mastery

For the rational function (2x⁴ + 15x³ + 25x² + 24x - 120)/(x(x + 3)(x - 2)(x² + 5)), which partial fraction decomposition is correct over the real numbers?

A4/x + (-2)/(x + 3) + 2/(x - 2) + (-3x + 6)/(x² + 5)

B(-4)/x + (-1)/(x + 3) + 2/(x - 2) + (-3x + 6)/(x² + 5)

C4/x + (-1)/(x + 3) + 2/(x - 2) + (-3x + 6)/(x² + 5)

D5/x + (-1)/(x + 3) + 2/(x - 2) + (-3x + 6)/(x² + 5)

Solution & Explanation:Clear the common denominator x(x + 3)(x - 2)(x² + 5). The decomposition 4/x + (-1)/(x + 3) + 2/(x - 2) + (-3x + 6)/(x² + 5) produces the numerator 2x⁴ + 15x³ + 25x² + 24x - 120, so it equals the given rational function. The irreducible quadratic factors need linear numerators, and any repeated factor needs a term for each power.

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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 full decomposition for a rational function with mixed linear and quadratic factors. Standards tags: 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).