Algebra II - Factoring Polynomials - Quick Guides | Mathematics

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Study GuideAlgebra II–Factoring Polynomials1. Greatest Common FactorWhat Does It Mean to Factor a Polynomial?Factoring a polynomial meansrewriting the polynomial as a product(a multiplication) of simplerexpressions such asmonomials or smaller polynomials.Because polynomials can appear in many different forms, there areseveral ways to factor them.One of the simplest and most common methods isfactoring out the Greatest Common Factor(GCF).What is the Greatest Common Factor?TheGreatest Common Factor (GCF)is thelargest factor that every term in the polynomial hasin common.When we factor out the GCF, we:1.Find the largest number or variable that dividesall the terms.2.Factor it out (place it outside parentheses).3.Write what remains inside the parentheses.Think of it likepulling out the common partshared by every term.Example 1:Factor:Step 1: Find the common factorBoth terms5xand5ycontain5.So, theGCF is 5.Step 2: Factor it out

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Study GuideWhy does this work?Divide each term by 5:•(5x ÷ 5 = x)•(5y ÷ 5 = y)So the expression becomes:Example 2:Factor:Step 1: Look for common factorsFirst check thenumbers:•24, 16, and 8 all share factors2, 4, and 8•Thegreatest common factor is 8Next check thevariables:•Each term containsxSo theGCF is (8x).Step 2: Factor out (8x)Step 3: Check by dividing each term•(24x3÷ 8x = 3x2)•(-16x2÷ 8x =-2x)•(8x ÷ 8x = 1)So the factorization is correct:

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Study GuideSummary•Factoringmeans rewriting a polynomial as aproduct of simpler expressions.•TheGreatest Common Factor (GCF)is thelargest factor shared by all terms.•To factor using the GCF:1.Find the greatest common number and variable in all terms.2.Factor it out.3.Write the remaining expression inside parentheses.Example Results•(5x + 5y = 5(x + y))•(24x3-16x2+ 8x = 8x(3x2-2x + 1))Key Idea:Always checkboth numbers and variableswhen finding the GCF.2. Quiz: Greatest Common Factor1. QuestionFactor the expression:9x³ + 27x²−18xAnswer Choices•9x(x² + 3x−2)•9x²(x + 3)−2•9(x³ + 3x²−2x)Correct Answer9x(x² + 3x−2)

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Study GuideWhy This Is CorrectFirst find theGreatest Common Factor (GCF)of all terms.Terms:9x³, 27x²,−18xCommon factors:•Numbers: GCF of9, 27, 18 = 9•Variables: smallest power ofx = xSo the GCF is9x.Factor it out:9x³ + 27x²−18x=9x(x² + 3x−2)2. QuestionFactor the expression:16x³y−32x²y² + 24xy³−16xyAnswer Choices•8x(2x²y−4xy² + 3y³−2y)•8y(2x³−4x²y + 3xy²−2x)•8xy(2x²−4xy + 3y²−2)Correct Answer8xy(2x²−4xy + 3y²−2)Why This Is CorrectFind theGCF.Numbers:GCF of16, 32, 24, 16 = 8

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Study GuideVariables:•smallestx = x¹•smallesty = y¹So the GCF is8xy.Factor it out:16x³y−32x²y² + 24xy³−16xy=8xy(2x²−4xy + 3y²−2)3. QuestionFactor the expression:7x⁵−28x⁴+ 35x³−14x²Answer Choices7x(x⁴−4x³ + 5x²−14x)7x²(x³−4x² + 5x−2)7x³(x²−4x + 5)−14Correct Answer7x²(x³−4x² + 5x−2)Why This Is CorrectFind theGCF.Numbers:GCF of7, 28, 35, 14 = 7Variables:Smallest power ofx = x²So the GCF is7x².

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Study GuideFactor:7x⁵−28x⁴+ 35x³−14x²=7x²(x³−4x² + 5x−2)4. QuestionFactor the expression:12a⁶+ 24a⁵b²−18a⁴b³ + 36a³b⁴−12a²b⁵Answer Choices•12a⁶+ 6a²b²(4a³−3a²b + 6ab²−2b³)•6a(2a⁵+ 4a⁴b²−3a³b³ + 6a²b⁴−2ab⁵)•6a²(2a⁴+ 4a³b²−3a²b³ + 6ab⁴−2b⁵)Correct Answer6a²(2a⁴+ 4a³b²−3a²b³ + 6ab⁴−2b⁵)Why This Is CorrectFind theGCF.Numbers:GCF of12, 24, 18, 36, 12 = 6Variables:Smallest power ofa = a²So the GCF is6a².Factor it out:12a⁶+ 24a⁵b²−18a⁴b³ + 36a³b⁴−12a²b⁵=6a²(2a⁴+ 4a³b²−3a²b³ + 6ab⁴−2b⁵)

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Study Guide3. Difference of Squares1. What is the Difference of Squares?In algebra, there is a special pattern called thedifference of squares. It happens whenone squareis subtracted from another square.A key rule to remember is:The expressions(a + b)and(a−b)are calledconjugates.When you multiply conjugates, the middle terms cancel, leaving adifference of squares.2.Example 1:Factor (x2-16)Notice that both terms areperfect squares.•(x2 =(x)2)•(16 = (4)2)So we can rewrite it as:Now apply the difference of squares rule:So the factored form is:3.Example 2:Factor (25x2y2-36z2)Again, check if both terms are perfect squares.•(25x2y2= (5xy)2)•(36z2 =(6z)2)

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Study GuideRewrite:Apply the difference of squares:4.Example 3:Factor ((a + b)2-(c-d)2)Both expressions are squares.Apply the formula:Simplify inside the parentheses:5.Example 4:Why (y2 +9) Cannot Be FactoredAlthough both terms are squares:•(y2 =(y)2)•(9 = (3)2)The expression is:This is asum of squares, not adifference of squares.Since the formula requires subtraction, this expressioncannot be factored using this method.

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Study Guide6. Using Multiple Factoring MethodsSometimes you need to usemore than one factoring method.A common strategy is:1.First:Factor out theGreatest Common Factor (GCF)2.Second:Check if the remaining expression is adifference of squares7.Example 5:Factor (9x2-36)First factor out theGCF (9):Now factor the difference of squares inside the parentheses:Final answer:8.Example 6:Factor (8(x + y)2-18)First factor out theGCF (2):Notice that:•(4(x + y)2 =[2(x + y)]2)•(9 = 32)Now apply the difference of squares:Simplify:

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Study GuideSummaryDifference of Squares RuleKey Points to Remember•It only works whentwo squares are subtracted.•Both terms must beperfect squares.•The result is a pair ofconjugate binomials.•Asum of squares(like (x2+9)) cannot be factored using this method.•Sometimes you mustfactor out the GCF first, then apply the difference of squares.Steps to Factor Using Difference of Squares1.Check if both terms areperfect squares.2.Make sure they arebeing subtracted.3.Use the formula:((a + b)(a-b)4. Quiz: Difference of Squares1. QuestionFactor the expression:9a²−49Answer Choices•(3a−7)(3a−7)•(3a + 7)(3a−7)•(3a + 7)(3a + 7)

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