温馨提示：本站仅提供公开网络链接索引服务，不存储、不篡改任何第三方内容，所有内容版权归原作者所有
AI智能索引来源：http://www.edu.com/math-glossary/Remainder-Theorem-Definition-Examples
点击访问原文链接

Remainder Theorem: Definition and Examples | EDU.COM

进入原网站 导航首页
Remainder Theorem: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Remainder TheoremRemainder Theorem: Definition and ExamplesTable of ContentsRemainder Theorem Definition of Remainder Theorem

The remainder theorem provides a shortcut for finding remainders when dividing polynomials. When we divide a polynomial p(xxx) by (x−a)(x − a)(x−a), the remainder equals p(aaa) - the value of the polynomial when x=ax = ax=a. So instead of doing long division, we can simply substitute x=ax = ax=a into the original polynomial to find the remainder.

The degree of the remainder polynomial is always 1 less than the degree of the divisor polynomial. When any polynomial is divided by a linear polynomial (polynomial with degree = 1), the remainder is always constant (degree = 0). Additionally, (x−a)(x − a)(x−a) is a divisor of the polynomial P(xxx) if and only if P(aaa) = 0, which is why this theorem is also used to factorize polynomials.

Examples of Remainder Theorem Example 1: Finding Remainder Using the Theorem Problem:

Find the remainder when p(x)=x2+4x+4p(x) = x^2 + 4x + 4p(x)=x2+4x+4 is divided by (x−1)(x − 1)(x−1).

Step-by-step solution:

Step 1, Find the value of aaa by setting the linear polynomial equal to zero.

x−1=0⇒x=1x - 1 = 0 \Rightarrow x = 1x−1=0⇒x=1 So, a=1a = 1a=1

Step 2, Use the remainder theorem formula. The remainder equals p(a)p(a)p(a).

Remainder = p(1)p(1)p(1)

Step 3, Substitute the value of aaa into the polynomial.

p(1)=12+4(1)+4p(1) = 1^2 + 4(1) + 4p(1)=12+4(1)+4

Step 4, Solve the expression.

p(1)=1+4+4=9p(1) = 1 + 4 + 4 = 9p(1)=1+4+4=9

Step 5, The remainder is 999.

Example 2: Finding Remainder with a Different Linear Divisor Problem:

Find the remainder when f(x)=2x2+4x−4f(x) = 2x^2 + 4x − 4f(x)=2x2+4x−4 is divided by (2x−1)(2x − 1)(2x−1).

Step-by-step solution:

Step 1, For the divisor (2x−1)(2x - 1)(2x−1), we need to find where it equals zero. 2x−1=0⇒x=122x - 1 = 0 \Rightarrow x = \frac{1}{2}2x−1=0⇒x=21​. Note: To apply the remainder theorem directly, we could also rewrite (2x−1)(2x - 1)(2x−1) as 2(x−12)2(x - \frac{1}{2})2(x−21​).

Step 2, Use the remainder theorem. The remainder equals f(a)f(a)f(a).

Remainder = f(12)f(\frac{1}{2})f(21​)

Step 3, Substitute x=12x = \frac{1}{2}x=21​ into the polynomial.

f(12)=2(12)2+4(12)−4f(\frac{1}{2}) = 2(\frac{1}{2})^2 + 4(\frac{1}{2}) - 4f(21​)=2(21​)2+4(21​)−4

Step 4, Simplify each term.

f(12)=2(14)+4(12)−4f(\frac{1}{2}) = 2(\frac{1}{4}) + 4(\frac{1}{2}) - 4f(21​)=2(41​)+4(21​)−4 f(12)=12+2−4f(\frac{1}{2}) = \frac{1}{2} + 2 - 4f(21​)=21​+2−4

Step 5, Calculate the final result.

f(12)=−32f(\frac{1}{2}) = \frac{-3}{2}f(21​)=2−3​

Step 6, The remainder is −32\frac{-3}{2}2−3​.

Example 3: Checking if a Binomial is a Factor of a Polynomial Problem:

Check if (x+3)(x + 3)(x+3) is a factor of x2+6x+9x^2 + 6x + 9x2+6x+9.

Step-by-step solution:

Step 1, Rewrite (x+3)(x + 3)(x+3) as (x−(−3))(x - (-3))(x−(−3)) to match the form (x−a)(x - a)(x−a).

So, a=−3a = -3a=−3

Step 2, According to the remainder theorem, (x−a)(x - a)(x−a) is a factor of a polynomial p(x)p(x)p(x) if and only if p(a)=0p(a) = 0p(a)=0.

Step 3, Calculate p(−3)p(-3)p(−3) by substituting x=−3x = -3x=−3 into the polynomial.

p(−3)=(−3)2+6(−3)+9p(-3) = (-3)^2 + 6(-3) + 9p(−3)=(−3)2+6(−3)+9

Step 4, Simplify the expression.

p(−3)=9−18+9=0p(-3) = 9 - 18 + 9 = 0p(−3)=9−18+9=0

Step 5, Since p(−3)=0p(-3) = 0p(−3)=0, we can say that (x+3)(x + 3)(x+3) is indeed a factor of x2+6x+9x^2 + 6x + 9x2+6x+9.

Comments(4)TTableTennisPlayerTheoNovember 6, 2025I've been struggling to explain the remainder theorem to my students. This page's clear def and examples made it so much easier! Thanks!

DDesignerMonaNovember 5, 2025I've used this remainder theorem def for my students. It's super clear! The examples really helped them grasp the concept fast.

WWebDeveloperXenaNovember 5, 2025This glossary page on the remainder theorem is great! It helped my students grasp the concept easily. Thanks for the clear def and examples!

MCMs. CarterSeptember 17, 2025This explanation of the Remainder Theorem was so clear! I used the examples to help my son with his algebra homework, and he finally got it. Thanks for the step-by-step breakdown—it’s a lifesaver for parents!

Explore More TermsPercent DifferenceEqual SignKilogramSquare NumbersVerticalTrapezoidView All Math TermsRecommended Interactive LessonsUnderstand Non-Unit Fractions Using Pizza Models3Math3.NF.A.1Use place value to multiply by 103Math3.OA.C.7Divide by 43Math3.OA.C.7Multiply by 53Math3.OA.C.7Multiply by 13Math3.OA.C.7Divide by 63Math3.OA.C.7View All Interactive LessonsRecommended VideosClassify and Count ObjectsKMathK.MD.B.3Read and Interpret Picture Graphs1Math1.MD.C.4Use models and the standard algorithm to divide two-digit numbers by one-digit numbers4Math4.NBT.B.6Prime And Composite Numbers4Math4.OA.B.4Find Angle Measures by Adding and Subtracting4Math4.MD.C.7Plot Points In All Four Quadrants of The Coordinate Plane6Math6.NS.C.6b, 6.NS.C.6cView All VideosRecommended WorksheetsClassify and Count ObjectsKMathK.MD.B.3Write Subtraction SentencesKMathK.OA.A.1Count by Ones and Tens1Math1.NBT.A.1Subtract Fractions With Like Denominators4Math4.NF.B.3aAdd, subtract, multiply, and divide multi-digit decimals fluently6Math6.NS.B.3Write Algebraic Expressions6Math6.EE.A.2aView All WorksheetsRecommended Coloring PagesSingle acacia tree in the center with a flat plain backgroundPre-K – KAll SubjectsElectric car with large wheels and smiling facePre-K – KAll SubjectsTropical fish with stripes swimming near a seaweed1 – 2All SubjectsDog shaking water off its fur in a simple backyard scene with grass1 – 2All SubjectsA meadow with wildflowers and a deer grazing3 – 4All SubjectsStegosaurus with a group of other dinosaurs in a prehistoric landscape5 – 6All SubjectsView All Coloring PagesRecommended BlogsABCya Sound Burst: A Teacher's Guide to Interactive Phonics LearningNovember 16, 2025Science Games Transform Middle School Learning: A Path to Better Engagement and UnderstandingNovember 16, 2025Fireboy and Watergirl Online Multiplayer: Fun Math Games Kids Can Play TogetherNovember 15, 2025Can You Make Your Dog Sit in Bloxd.io? A Parent's Guide to Gaming and Real-World LearningNovember 15, 2025Runaway Toad: The Perfect Math Game for Your ClassroomNovember 15, 2025Suika Watermelon Game: 5 Math Strategies for K-6 ClassroomsNovember 14, 2025View All Blog PostsQUICK LINKSAbout UsPrivacy PolicyTerms of ServiceTOOLSHomework HelperGuide DesignerPodcast MakerPlan BuilderRESOURCESMath GlossaryEnglish GlossaryEnglish Language ArtsMathematicsScienceBook InsightsFun with WordsBlog© 2025 EDU.COM. All rights reserved.

智能索引记录