Dividing by a Quadratic
Part of the Polynomials & Rational Expressions section of Coddy's Algebra 2 journey. Lesson 5 of 54.
Nothing changes when the divisor is a quadratic: you still divide leading terms. Take (x^3 + 3x^2 + 4x + 5)/(x^2 + x + 1). x^3 over x^2 is x. Multiply: x(x^2 + x + 1) is x^3 + x^2 + x. Subtract: 2x^2 + 3x + 5 is left.
Second round: 2x^2 over x^2 is 2. Multiply: 2(x^2 + x + 1) is 2x^2 + 2x + 2. Subtract: x + 3 is left. That still has an x, but its degree, 1, is lower than the divisor's, 2, so you stop:
(x^3 + 3x^2 + 4x + 5)/(x^2 + x + 1) = x + 2 + (x + 3)/(x^2 + x + 1)
So with a quadratic divisor the remainder can hold an x. The stopping rule is the one you know: the remainder's degree must be lower than the divisor's. A divisor with a gap needs the same care as a top with one: line up like terms before you subtract.
Challenge
Divide x^3 + 3x^2 - x + 2 by x^2 + 1. Stop when the remainder's degree is lower than the divisor's.
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