A Remainder Left Over
Part of the Polynomials & Rational Expressions section of Coddy's Algebra 2 journey. Lesson 3 of 54.
Most divisions do not come out exact. Take (2x^2 + 3x + 4)/(x + 1). First round: 2x^2 over x is 2x, and 2x(x + 1) is 2x^2 + 2x. Subtracting leaves x + 4, so the board shows
2x + (x + 4)/(x + 1)
Second round: x over x is 1, and subtracting x + 1 leaves 3. Now no x is left, so the division stops. 3 is the remainder, and it stays over the divisor:
(2x^2 + 3x + 4)/(x + 1) = 2x + 1 + 3/(x + 1)
Why stop there? Going on would mean dividing 3 by x, which is not a polynomial term. The rule: stop when the remainder has a lower degree than the divisor. To check, multiply back: (x + 1)(2x + 1) + 3 is 2x^2 + 3x + 1 + 3, the original top.
Challenge
Divide 2x^2 + 5x - 1 by x + 3. Keep pressing Long division by x + 3 until what is left is a plain number.
Try it yourself
Divide the polynomials
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