Sum of Cubes
Part of the Polynomials & Rational Expressions section of Coddy's Algebra 2 journey. Lesson 15 of 54.
A sum of two cubes factors too, with the signs moved around:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
Compare it with the difference: the short bracket has the same sign as the cubes, the middle of the long bracket has the opposite sign, and the last term is always plus. The cubes can carry a number. In 27x^3 + 8, 27x^3 is (3x)^3 and 8 is 2 cubed, so a = 3x and b = 2. Then a squared is 9x^2 and ab is 6x:
27x^3 + 8 = (3x + 2)(9x^2 - 6x + 4)
Unlike a sum of squares, a sum of cubes always has a real linear factor: a^3 + b^3 is zero when a = -b, so a + b is a factor. Multiply out to check: -18x^2 and 18x^2 cancel, and so do -12x and 12x.
Challenge
Factor 8x^3 + 1. Type the factored form as the next line; the board accepts it only if it equals the start.
Try it yourself
Factor 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 Charts3Factoring Higher Degree
Factor by GroupingGrouping All the WayDifference of CubesSum of CubesA Quadratic in DisguiseRecap: Factoring6Rational Expressions
Cancelling FactorsExcluded ValuesMultiplyingDividingRecap: Rational Expressions