Common Factor First
Part of the Polynomials & Rational Expressions section of Coddy's Algebra 2 journey. Lesson 18 of 54.
An equation like x^3 = 25x tempts you to divide both sides by x. That gives x^2 = 25, so x is 5 or -5. But 0 works too: 0 cubed is 0, and 25 times 0 is 0. Dividing by x threw that answer away, because dividing by x is only allowed when x is not 0.
The safe route is to get 0 on one side and factor. In x^3 - 25x = 0 both terms hold x, so take it out: x(x^2 - 25) = 0. A product is 0 only when one of its factors is 0, so split into cases: x = 0, or x^2 - 25 = 0, which gives 5 and -5. That makes three answers, and a cubic can have up to three.
Challenge
Solve x^3 - 9x = 0. Find every solution, without dividing by x.
Try it yourself
Solve for x
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