The Remainder Theorem
Part of the Polynomials & Rational Expressions section of Coddy's Algebra 2 journey. Lesson 7 of 54.
Divide p(x) = x^2 + 3x - 4 by x - 2 and you get quotient x + 5 with remainder 6. Now put 2 into p: p(2) is 4 + 6 - 4, which is also 6. That is no coincidence.
Any division can be written as divisor times quotient plus remainder. Here that reads
p(x) = (x - 2)(x + 5) + 6
Put x = 2 into both sides. The bracket x - 2 becomes zero, so the whole product is zero, and what is left is the remainder. In general, the remainder theorem says: the remainder of p divided by x - a is p(a). So you can find a remainder with no division at all. Watch the sign: dividing by x + 5 means x - (-5), so you evaluate p(-5).
Challenge
Find the remainder when p(x) = x^3 - 4x^2 + 2x + 5 is divided by x - 3, without dividing. The board holds p(3). Tap Use p(x) = ..., then type the arithmetic line by line.
Try it yourself
Simplify the expression
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Polynomials & Rational Expressions
1Polynomial Long Division
How Long Division WorksOne Term at a TimeA Remainder Left OverA Missing TermDividing by a QuadraticRecap: Long Division7Adding and Subtracting
Same BottomDifferent BottomsSubtractingFactor the Bottoms FirstRecap: Adding Fractions2Remainders and Factors
The Remainder TheoremThe Factor TheoremFinding a Missing NumberUsing a Known FactorRecap: Remainders and Factors5Sign Charts
Between the RootsOutside the RootsA Negative Leading TermReading a Sign ChartRecap: Sign Charts