Polynomial Division

Explore long division, synthetic division, quotient-remainder identity.

Lesson 2 of 53 parts Practice Flashcards

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Long Division

Objective: Explain long division and use it to solve or interpret a problem.

Polynomial long division repeats leading-term division, multiplication, subtraction, and bringing down until the remainder has lower degree than divisor. A remainder whose degree is below the divisor degree.

Worked examples

Try itPolynomial Division — practice 1.1: Divide x²+7x+10 by x+2. What is the constant of the quotient?Show answer

Step 1: Factor the numerator as (x+2)(x+5). Step 2: Cancel x+2 in polynomial division. Step 3: The quotient is x+5; constant 5.

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Long Division

Objective: Explain long division and use it to solve or interpret a problem.

Worked examples

Try itPolynomial Division — practice 1.1: Divide x²+7x+10 by x+2. What is the constant of the quotient?Show answer

Step 1: Factor the numerator as (x+2)(x+5). Step 2: Cancel x+2 in polynomial division. Step 3: The quotient is x+5; constant 5.

Synthetic Division

Objective: Explain synthetic division and use it to solve or interpret a problem.

Synthetic division is a compact method for dividing a polynomial by x−r; write missing-power coefficients as zeros. They keep powers correctly aligned.

Worked examples

Try itPolynomial Division — practice 2.1: Find the remainder when p(x)=x²+3x+7 is divided by x−8.Show answer

Step 1: Use the Remainder Theorem: remainder=p(8). Step 2: Compute 8²+3(8)+7. Step 3: The result is 95.

Quotient-Remainder Identity

Objective: Explain quotient-remainder identity and use it to solve or interpret a problem.

For divisor d(x)≠0, p(x)=d(x)q(x)+r(x), where deg r<deg d. Multiply and add back to check the division. Multiply divisor by quotient and add the remainder.

Worked examples

Try itPolynomial Division — practice 3.1: Divide x²+15x+44 by x+4. What is the constant of the quotient?Show answer

Step 1: Factor the numerator as (x+4)(x+11). Step 2: Cancel x+4 in polynomial division. Step 3: The quotient is x+11; constant 11.

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