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3.3 Dividing Polynomials

Use the remainder theorem.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Perform long division and state both division identitiesWorked steps · five questions02Divide by nonmonic linear divisorsWorked steps · five questions03Divide by quadratic divisors with missing powersWorked steps · five questions04Perform synthetic division with signed or fractional inputsWorked steps · five questions05Evaluate with the remainder theorem and direct substitutionWorked steps · five questions06Use a known zero to factor and find all other zerosWorked steps · five questions07Construct a polynomial from roots and a coefficientWorked steps · five questions08Construct integer-coefficient polynomials with irrational rootsWorked steps · five questions09Recover polynomials from graphed roots and an additional pointWorked steps · five questions10Use remainder and factor shortcuts for very high degreesWorked steps · five questions11Prove and use Horner formWorked steps · five questions12Prove the remainder theorem and both factor-theorem directionsWorked steps · five questions
Lesson brief and rules

3.3 Dividing Polynomials

When dividing p(x)p(x) by x−ax - a, the remainder is p(a)p(a). A remainder of zero means x−ax - a is a factor.

Remember
Use a, not −a, when the divisor is x − a.
Worked example 11
01 / Use long division on x3+x2−3x+10x^{3} + x^{2} - 3 x + 10 divided by x+3x + 3. Identify dividend, divisor, quotient and remainder; write both division identities and the domain restriction for the fraction. Show your work; a teacher must review the construction or explanation. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
02 / Arrange both polynomials in descending powers; retain a zero coefficient for each missing power.
03 / Divide the leading terms to obtain x2x^{2}.
04 / Multiply the divisor: x3+3x2x^{3} + 3 x^{2}. Subtract from x3+x2−3x+10x^{3} + x^{2} - 3 x + 10 to get −2x2−3x+10- 2 x^{2} - 3 x + 10.
05 / Divide the leading terms to obtain −2x- 2 x.
06 / Multiply the divisor: −2x2−6x- 2 x^{2} - 6 x. Subtract from −2x2−3x+10- 2 x^{2} - 3 x + 10 to get 3x+103 x + 10.
07 / Divide the leading terms to obtain 33.
08 / Multiply the divisor: 3x+93 x + 9. Subtract from 3x+103 x + 10 to get 11.
09 / The remainder is zero or has degree smaller than the divisor.
10 / Check by expansion: x3+x2−3x+10=(x+3)(x2−2x+3)+(1)x^{3} + x^{2} - 3 x + 10=(x + 3)(x^{2} - 2 x + 3)+(1).
11 / The polynomial identity holds at every real input; the fractional identity excludes zeros of the divisor.
Worked example 2 · Perform long division and state both division identities

Use long division on 2x3+x2−3x+82 x^{3} + x^{2} - 3 x + 8 divided by x+2x + 2. Identify dividend, divisor, quotient and remainder; write both division identities and the domain restriction for the fraction. Show your work; a teacher must review the construction or explanation. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

Arrange both polynomials in descending powers; retain a zero coefficient for each missing power.
Divide the leading terms to obtain 2x22 x^{2}.
Multiply the divisor: 2x3+4x22 x^{3} + 4 x^{2}. Subtract from 2x3+x2−3x+82 x^{3} + x^{2} - 3 x + 8 to get −3x2−3x+8- 3 x^{2} - 3 x + 8.
Divide the leading terms to obtain −3x- 3 x.
Multiply the divisor: −3x2−6x- 3 x^{2} - 6 x. Subtract from −3x2−3x+8- 3 x^{2} - 3 x + 8 to get 3x+83 x + 8.
Divide the leading terms to obtain 33.
Multiply the divisor: 3x+63 x + 6. Subtract from 3x+83 x + 8 to get 22.
The remainder is zero or has degree smaller than the divisor.
Check by expansion: 2x3+x2−3x+8=(x+2)(2x2−3x+3)+(2)2 x^{3} + x^{2} - 3 x + 8=(x + 2)(2 x^{2} - 3 x + 3)+(2).
The polynomial identity holds at every real input; the fractional identity excludes zeros of the divisor.
Written-work review:

Read the graphs

Recover polynomials from graphed roots and an additional point

The displayed graph is a polynomial of degree 33. The marked yy-intercept is (0,6)(0,6). Read its zeros and crossing and touching behavior and recover its polynomial.
Graph of PAxes labeled x and y. Window x from -3 to 4, y from -33.2 to 25.400000000000002. Open circles exclude endpoints; filled circles include them.-3-2-101234-32-24-16-8081624P: solidxy
Show your work; a teacher must review the construction or explanation. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

The visible zeros are −2-2, 11, 33.
The root 11 has odd multiplicity at least 11; the other two cross.
The specified degree leaves exactly the displayed minimal multiplicities.
Substitute the marked yy-intercept into the product form; the scale factor is 11.
Written-work review:

  • Show the requested algebra or numerical evidence, not merely a selected option.
  • Include all requested coordinates, domains, units, comparisons and conclusions.
  • For a graph, label axes, intercepts and extrema; use correct end behavior and a suitable scale.

Answer: P(x)=(x−3)(x−1)(x+2)P(x)=\left(x - 3\right) \left(x - 1\right) \left(x + 2\right)

15 questions with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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