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

CAL2.35 Rational Functions and Polynomial Division Worksheet

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Q: What 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 quotient and remainder before integrating an improper rational function. 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.).

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.35 Rational Functions and Polynomial Division. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.35 Rational Functions and Polynomial Division - Select the correct quotient and remainder before integrating an improper rational function. 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 CAL2.35 Rational Functions and Polynomial Division skill assessments:

Problem 1Difficulty: Mastery

A student must divide P(x)=-3x⁸ + 7x⁷ + 12x⁶ - 22x⁵ - 6x⁴ - 14x³ + 15x² - 4x by D(x)=x⁴ - 4x² - 2x - 3 before integrating P(x)/D(x). Which pair is correct with deg r < deg D?

Aq(x)=-3x⁴ + 7x³ - 1, r(x)=-7x³ - 11x² + 6x + 3

Bq(x)=-3x⁴ + 7x³, r(x)=-x⁴ + 7x³ + 15x² - 4x

Cq(x)=-3x⁴ + 7x³ - 1, r(x)=7x³ + 11x² - 6x - 3

Dq(x)=-3x⁴ + 7x³ - 2, r(x)=x⁴ + 7x³ + 7x² - 8x - 6

Solution & Explanation:Polynomial division gives q(x)=-3x⁴ + 7x³ - 1 and r(x)=7x³ + 11x² - 6x - 3 because (x⁴ - 4x² - 2x - 3)(-3x⁴ + 7x³ - 1)+(7x³ + 11x² - 6x - 3) expands to -3x⁸ + 7x⁷ + 12x⁶ - 22x⁵ - 6x⁴ - 14x³ + 15x² - 4x. The remainder has degree 3, which is less than the divisor degree 4, so the integrand becomes -3x⁴ + 7x³ - 1 + (7x³ + 11x² - 6x - 3)/(x⁴ - 4x² - 2x - 3). A common mistake is to stop with an algebraic identity whose remainder still has the same degree as the divisor, or to reverse the remainder's sign.

Problem 2Difficulty: Mastery

To split ∫ [(9x⁹ - 15x⁸ + 30x⁷ - x⁶ - 25x⁵ + 30x⁴ - 16x³ + 8x² - x - 1)/(3x⁴ - 2x³ + 1)] dx into a polynomial part plus a proper rational part, choose q(x) and r(x) with P=Dq+r and deg r

Aq(x)=3x⁵ - 3x⁴ + 8x³ + 5x² - 6x + 7, r(x)=-10x³ + 3x² + 5x - 8

Bq(x)=3x⁵ - 3x⁴ + 8x³ + 5x² - 6x + 6, r(x)=3x⁴ - 12x³ + 3x² + 5x - 7

Cq(x)=3x⁵ - 3x⁴ + 8x³ + 5x² - 6x + 8, r(x)=-3x⁴ - 8x³ + 3x² + 5x - 9

Dq(x)=3x⁵ - 3x⁴ + 8x³ + 5x² - 6x + 7, r(x)=10x³ - 3x² - 5x + 8

Solution & Explanation:Polynomial division gives q(x)=3x⁵ - 3x⁴ + 8x³ + 5x² - 6x + 7 and r(x)=-10x³ + 3x² + 5x - 8 because (3x⁴ - 2x³ + 1)(3x⁵ - 3x⁴ + 8x³ + 5x² - 6x + 7)+(-10x³ + 3x² + 5x - 8) expands to 9x⁹ - 15x⁸ + 30x⁷ - x⁶ - 25x⁵ + 30x⁴ - 16x³ + 8x² - x - 1. The remainder has degree 3, which is less than the divisor degree 4, so the integrand becomes 3x⁵ - 3x⁴ + 8x³ + 5x² - 6x + 7 + (-10x³ + 3x² + 5x - 8)/(3x⁴ - 2x³ + 1). A common mistake is to stop with an algebraic identity whose remainder still has the same degree as the divisor, or to reverse the remainder's sign.

Problem 3Difficulty: Mastery

The rational function (8x⁸ - 8x⁷ - 20x⁶ + 8x⁵ + 2x⁴ - 16x³ + 4x² + 39x - 23)/(2x³ - 2x² - 5x + 2) is improper. Which quotient-remainder pair with deg r < deg D should be used before integration?

Aq(x)=4x⁵ + x - 8, r(x)=2x³ - 7x² - 3x - 7

Bq(x)=4x⁵ + x - 7, r(x)=-5x² + 2x - 9

Cq(x)=4x⁵ + x - 6, r(x)=-2x³ - 3x² + 7x - 11

Dq(x)=4x⁵ + x - 7, r(x)=5x² - 2x + 9

Solution & Explanation:Polynomial division gives q(x)=4x⁵ + x - 7 and r(x)=-5x² + 2x - 9 because (2x³ - 2x² - 5x + 2)(4x⁵ + x - 7)+(-5x² + 2x - 9) expands to 8x⁸ - 8x⁷ - 20x⁶ + 8x⁵ + 2x⁴ - 16x³ + 4x² + 39x - 23. The remainder has degree 2, which is less than the divisor degree 3, so the integrand becomes 4x⁵ + x - 7 + (-5x² + 2x - 9)/(2x³ - 2x² - 5x + 2). A common mistake is to stop with an algebraic identity whose remainder still has the same degree as the divisor, or to reverse the remainder's sign.

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Frequently Asked Questions

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QHow do I download and print this CAL2.35 Rational Functions and Polynomial Division worksheet?

Click either the "Download PDF" or "Download DOCX" button at the top of the page. You can print the PDF directly from your browser or edit the DOCX file in Microsoft Word or Google Docs to tailor the questions to your students.

QIs this CAL2.35 Rational Functions and Polynomial Division worksheet free for teachers and homeschoolers?

Yes! All learning materials, worksheets, and practice resources on Free K12 Practice are 100% free for teachers, students, parents, tutors, and homeschooling families for non-commercial educational use. No account or subscription is required.

QCan students practice CAL2.35 Rational Functions and Polynomial Division online with instant feedback?

Yes! In addition to this printable worksheet, students can practice CAL2.35 Rational Functions and Polynomial Division interactively on our platform with automated grading, instant score tracking, and detailed step-by-step solutions.

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 quotient and remainder before integrating an improper rational function. 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.).