A Difference of Squares
Part of the Exponents & Polynomials section of Coddy's Algebra journey. Lesson 52 of 64.
One special pair remains: a sum times a difference of the same two things. (x+3)(x-3): the four products are x^2, -3x, 3x, -9, and the cross products are opposite twins. They cancel to nothing.
What survives is short: x^2 - 9, a difference of squares. In letters: (a+b)(a-b) = a^2 - b^2. This is the one shape whose product genuinely has no middle term, which is what makes it special enough to name.
It even does mental arithmetic: 21 * 19 is (20+1)(20-1) = 400 - 1 = 399. Recognizing the shape, in both directions, pays off through the whole next section.
Challenge
MediumExpand (x+6)(x-6) and watch the middle disappear.
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 Exponents & Polynomials
1What a Power Is
Repeated MultiplicationSquares and CubesPowers of Negative NumbersWhere the Minus LivesRecap: First Powers4Power of a Power
A Power Raised AgainWhy Exponents MultiplyProduct Rule or Power Rule?The Rule in LettersRecap: Power of a Power7Scientific Notation
Very Large NumbersBack to Standard FormMore Practice, Bigger NumbersVery Small NumbersMultiplying the NotationRecap: Scientific Notation10Multiplying Polynomials
Monomial Times PolynomialMinding the SignsBinomial Times BinomialWith CoefficientsThe Square of a BinomialA Difference of SquaresRecap: Multiplying2The Product Rule
Multiplying Same BasesWith CoefficientsMore Than Two FactorsThe Rule in LettersRecap: The Product Rule5Product and Quotient Powers
Every Factor Gets the PowerWith a Number InsideA Power of a QuotientAll Three Rules TogetherRecap: Products and Quotients8What a Polynomial Is
Terms with PowersThe DegreeLike Terms, Only StrongerRecap: Meet the Polynomials11Dividing by a Monomial
Splitting the FractionDividing by a Variable TermBigger SplitsCancel with CareRecap: Dividing3The Quotient Rule
Dividing Same BasesWhy Exponents SubtractWith CoefficientsThe Rule in LettersRecap: The Quotient Rule6Zero and Negative Exponents
The Zero ExponentZero at WorkNegative Means ReciprocalWriting Without NegativesPowers of TenRecap: Zero and Negative