With Coefficients
Part of the Exponents & Polynomials section of Coddy's Algebra journey. Lesson 50 of 64.
Coefficients and signs change the arithmetic, never the choreography. (2x+1)(x-3): four products, each one a small exercise from earlier chapters. 2x * x = 2x^2; 2x * (-3) = -6x; 1 * x = x; 1 * (-3) = -3.
Collect the x terms, signs attached: -6x + x = -5x. The product is 2x^2 - 5x - 3.
Notice what stopped being true: with a leading coefficient, the middle is no longer just "sum of the constants". The cross products each carry a coefficient now, and only writing all four products keeps the bookkeeping honest.
Challenge
MediumExpand (3x+2)(x-1) and collect.
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