Algebra II - Polynomial Arithmetic | Mathematics

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Study GuideAlgebra II–Polynomial ArithmeƟc1. Adding and SubtracƟng PolynomialsPolynomials are expressions made of one or moreterms.Each term is separated by aplus (+)orminus (−)sign.In a polynomial:•The exponents on variables must bewhole numbers(0, 1, 2, 3, …).•There isno limitto how long a polynomial can be.Some operations with polynomials are simple, but others require special methods.To add or subtract polynomials successfully, you need to understand:•Monomials, binomials, and trinomials•Like terms•Ascending and descending orderMonomials, Binomials, and TrinomialsMonomialAmonomialis a single term.•a number•a variable•or a number multiplied by variables.Rules for MonomialsFor an expression to be a monomial:1.Variables must havewhole number exponents.2.Variablescannot be inside square roots.3.Variablescannot appear in denominators.

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Study GuideExamples of Monomials•−12•a•3t²•(5/8)x2y3)•(y3)•(2(x+y2)/3)Expressions That Are NOT MonomialsThe following arenot monomials:•(x-2)→exponent isnot a whole number•(√x)→variable isinside a radical•(3(x+2)/(x+3))→variable appears in thedenominatorBinomialAbinomialis thesum of two monomials.Example:TrinomialAtrinomialis thesum of three monomials.Example:

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Study GuidePolynomialApolynomialcan be:•amonomial•thesum of two or more monomialsExamples:•(5x2)•(x2+ 3x + 2)Like TermsLike termsare terms that have thesame variables with the same exponents.Example of like terms:They are like terms because each contains(x2y).Not Like TermsThese arenot like terms:Their variable parts are different.Adding Like TermsTo add like terms:1.Add the numerical coefficients.2.Keep the variable part the same.

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Study GuideExample:If the terms arenot like terms, theycannot be combined.Example 1:Find the Sums1.(4x2y + 8x2y)= (12x2y)2.(-9abc + 3abc)= (-6abc)3.(9xy + 7x-28xy-4x)Group like terms:These two termscannot be combined furtherbecause they are unlike terms.Ascending and Descending OrderWhen writing polynomials with one variable, we often arrange them by thesize of the exponents.Descending OrderThe exponents gofrom largest to smallest.Example:

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Study GuideAscending OrderThe exponents gofrom smallest to largest.Example:Example 2:Write in Descending Powers of xOriginal expression:Arrange by powers ofx:Adding PolynomialsTo add polynomials:1.Combine like terms.2.Write the result in order(usually descending order).Given expression:Step 1: Identify the Powers of (x)Arrange the terms based on thepower of (x)from highest to lowest.

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Study GuideStep 2: Arrange in Descending Order of (x)Start with the highest power of (x) and move down:Final AnswerExample 3:Add the PolynomialsFirst combine like terms.(x3) terms(x2) terms(x) terms

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Study GuideconstantsFinal answer:Vertical MethodPolynomials can also be addedvertically, just like numbers.Write them indescending orderand line uplike terms:Subtracting PolynomialsTo subtract polynomials,add the oppositeof the polynomial being subtracted.Example 4:SubtractfromWrite the expression:

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Study GuideChange signs of the second polynomial:Combine like terms:SummaryKey Ideas• Apolynomialis made of one or more terms.• Terms are separated by+ or−signs.Types of Expressions•Monomial:one term•Binomial:two terms•Trinomial:three termsMonomial Rules•Exponents must bewhole numbers•No variables underradicals•No variables indenominatorsLike TermsTerms are like terms if they have:•thesame variables•thesame exponentsAdding Like Terms•Add thecoefficients•Keep thevariable part the same

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Study GuidePolynomial Order•Descending order:largest exponent→smallest•Ascending order:smallest exponent→largestAdding Polynomials1.Combinelike terms2.Write the result inorderSubtracting Polynomials•Add the oppositeof the polynomial being subtracted.2. Quiz: Adding and SubtracƟng Polynomials1. QuestionWhich of the following is themonomial?Answer Choices•(35xyz²) / 7•(16xyz²) / (2q)•35x + 21Correct Answer(35xyz²) / 7Why This Is CorrectAmonomialis a single algebraic term that may contain numbers and variables with non-negativeexponents.((35xyz²)/7) simplifies to5xyz², which is one single term.The other options are not monomials because:•((16xyz²)/(2q)) contains a variable in the denominator.•35x + 21has two terms (a binomial).

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Study Guide2. QuestionWhich of the following is thetrinomial?Answer Choices•(2x−13)•(21x + 7yx²−13) / 7•(2x + 7y²−13z) / 3xCorrect Answer(21x + 7yx²−13) / 7Why This Is CorrectAtrinomialhas exactlythree terms.The expression ((21x + 7yx²−13)/7) containsthree termsin the numerator.The others are not trinomials because:•2x−13has only two terms (binomial).•((2x + 7y²−13z)/(3x)) has variables in the denominator.3. QuestionWhich of the following is thebinomial?Answer Choices•2x−13/y•3x² + 5x−13•2x−13

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