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ZingPath: Polynomial Operations

The Remainder Theorem

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Lesson Focus

The Remainder Theorem


Learning Made Easy

Learners use the remainder theorem to find the remainder when dividing a polynomial P(x) by a divisor D(x) =<img src../../../tutorials/images/teacherguides/US820412IE_1.png" width="70" height="28" alt="" border="0">

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Find the remainder of a polynomial when it is divided by a linear polynomial.
  • Find the remainder of a polynomial when it is divided by a polynomial that is in the form <img src../../../tutorials/images/teacherguides/US820412IE_2.png\" width=\"66\" height=\"28\" alt=\"\" border=\"0\">

Everything You'll Have Covered

Polynomial long division is a method of dividing one polynomial by another nonzero polynomial that has lesser degree. It is a generalization of the familiar long division algorithm, which computes the quotient and remainder corresponding to the division of a dividend by some nonzero divisor.

Let be polynomials such that the degree of D(x) is less than the degree of P(x) and There are polynomials Q(x) and R(x) such that P(x)=Q(x) . D(x) + R(x); P(x) is the dividend, D(x) the divisor, Q(x) the quotient, and R(x) the remainder.

For example, let To compute P(x)/D(x), use the following procedure:

The quotient and remainder are Q(x) = x+4 and R(x) = 4, respectively. It is straightforward to verify that

The remainder theorem, sometimes called little Bézout's theorem, states that the remainder of a polynomial P(x) when divided by a linear polynomial (ax+b) is equal to Note that is the zero of (ax + b). In the case that , so the remainder is simply P(-b). The Activity Object introduces the remainder theorem by using this special case.

Tutorial Details

Approximate Time 30 Minutes
Pre-requisite Concepts Concept of polynomials, use polynomial long division to divide polynomials
Course Algebra-2
Type of Tutorial Skills Application
Key Vocabulary linear polynomial, polynomial, remainder