Polynomials - Math is Fun

that can be combined using addition, subtraction, multiplication and division ... ... So: A polynomial can have constants, variables and exponents, but never ... Polynomials Apolynomiallookslikethis: exampleofapolynomial thisonehas3terms Polynomialcomesfrompoly-(meaning"many")and-nomial(inthiscasemeaning"term")...soitsays"manyterms" Polynomialswithonevariablemakenicesmoothcurves: Apolynomialcanhave: constants(like3,−20,or½) variables(likexandy) exponents(likethe2iny2),butonly0,1,2,3,...etcareallowed thatcanbecombinedusingaddition,subtraction,multiplicationanddivision... ...except... ...notdivisionbyavariable(sosomethinglike2/xisrightout) So: Apolynomialcanhaveconstants,variablesandexponents, butneverdivisionbyavariable. Alsotheycanhaveoneormoreterms,butnotaninfinitenumberofterms. PolynomialorNot? Thesearepolynomials: 3x x−2 −6y2−( 7 9 )x 3xyz+3xy2z−0.1xz−200y+0.5 512v5+99w5 5 (Yes,"5"isapolynomial,onetermisallowed,anditcanbejustaconstant!)   Thesearenotpolynomials 3xy-2isnot,becausetheexponentis"-2"(exponentscanonlybe0,1,2,...) 2/(x+2)isnot,becausedividingbyavariableisnotallowed 1/xisnoteither √xisnot,becausetheexponentis"½"(seefractionalexponents)   Buttheseareallowed: x/2isallowed,becauseyoucandividebyaconstant also3x/8forthesamereason √2isallowed,becauseitisaconstant(=1.4142...etc) Monomial,Binomial,Trinomial Therearespecialnamesforpolynomialswith1,2or3terms: Howdoyourememberthenames?Thinkcycles! Thereisalsoquadrinomial(4terms)andquintinomial(5terms), butthosenamesarenotoftenused. Variables Polynomialscanhavenovariableatall Example:21isapolynomial.Ithasjustoneterm,whichisaconstant. Oronevariable Example:x4−2x2+x hasthreeterms,butonlyonevariable(x) Ortwoormorevariables Example:xy4−5x2z hastwoterms,andthreevariables(x,yandz) WhatisSpecialAboutPolynomials? Becauseofthestrictdefinition,polynomialsareeasytoworkwith. Forexampleweknowthat: Ifyouaddpolynomialsyougetapolynomial Ifyoumultiplypolynomialsyougetapolynomial Soyoucandolotsofadditionsandmultiplications,andstillhaveapolynomialastheresult. Also,polynomialsofonevariableareeasytograph,astheyhavesmoothandcontinuouslines. Example:x4−2x2+x Seehowniceand smooththecurveis? Youcanalsodividepolynomials(buttheresultmaynotbeapolynomial). Degree Thedegreeofapolynomialwithonlyonevariableisthelargestexponentofthatvariable. Example: 4x3−x+2 TheDegreeis3(thelargestexponentofx) Formorecomplicatedcases,readDegree(ofanExpression). StandardForm TheStandardFormforwritingapolynomialistoputthetermswiththehighestdegreefirst. Example:PutthisinStandardForm:3x2−7+4x3+x6 Thehighestdegreeis6,sothatgoesfirst,then3,2andthentheconstantlast: x6+4x3+3x2−7 Youdon'thavetouseStandardForm,butithelps.   342,343,1095,1096,3182,3183,3184,3185,1097,4002 AddingPolynomials IntroductiontoAlgebra MultiplyingPolynomials Algebra-BasicDefinitions AlgebraIndex Copyright©2021MathsIsFun.com