# The prime 5 does not appear in all factorizations

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The prime 5 does not appear in all factorizations: leave it out of the GCF
Stanley OckenM19500 Precalculus Chapter 1.3: Algebraic expressions
WelcomeAlgebraic ExpressionsSimplified sumsGreatest common factorFactoringExercisesQuiz ReviewExample 14:Find the GCF of81and64.Here81 = 34and64 = 26There isn’t any prime that appears in all factorizations andso the GCF is1.Example 15:Find the GCF of360,450,and1500.Step 1:360 = 23·32·51450 = 2·33·521500 = 22·3·53Step 2: The prime2appears in all factorizations. The lowest power is21= 2(in450) .The prime 3 appears in all factorizations: The lowest power is31= 3(in1500) .The prime 5 does not appear in all factorizations: The lowest power is51= 5(in360) .Answer:The GCF is the product2·3·5 = 30.Check:360is a common factor since360 = 30·12,450 = 30·15,and1500 = 30·50.Further check: The remaining factors12 = 22·3,15 = 3·5,and502·52have nocommon factor.Stanley OckenM19500 Precalculus Chapter 1.3: Algebraic expressions
WelcomeAlgebraic ExpressionsSimplified sumsGreatest common factorFactoringExercisesQuiz ReviewThe same idea applies to monomials:Example 16:Find the GCF of the 3 monomialsx3y4,x2y6z2,andxy8z4Solution:The primes that appear arex, y, zxappears in all monomials, with lowest powerx1=xyappears in all monomials, with lowest powery4zdoesn’t appear in all monomials, leave it out of the GCFAnswer:The GCF is the product of lowest powers:xy4.Sometimes we will phrase the solution method slightly differently:Solution (version 2):The 3 monomials can be rewrittenx3y4z0,x2y6z2,andx1y8z4. The GCF is the product of lowest powers:x1y4z0=xy4.Stanley OckenM19500 Precalculus Chapter 1.3: Algebraic expressions
WelcomeAlgebraic ExpressionsSimplified sumsGreatest common factorFactoringExercisesQuiz ReviewFactoring polynomialsProcedureTofactor a polynomialP:RewritePin standard form as a sum of monomials;Find the GCF of the coefficients and factor it out of the sum;Then find the GCF of the variable parts and factor it out.Example 17:Factor10x5y2+ 20x2y3z+ 30x7y5z3The GCF of the coefficients10,20,30is10. Factor it out to get10(x5y2+ 2x2y3z+ 3x7y5z3)The GCF of the variable partsx5y2, x2y3z,andx7y5z3isx2y2. Factor it out:10x2y2(x3+ 2yz+ 3x5y3z3)Additional Factoring Examples:2x5+ 3x2=x2(2x3+ 3)We factored out the lowest power of x2x5+ 3x2+ 7can’t be factored further, sincexis missing from the last term.x3y5+x2y8=x2y5(x+y3)We factored out lowest powers ofxandy.x3y5+x2y8+x3y4z3=x2y4(xy+y4+xz3)Stanley OckenM19500 Precalculus Chapter 1.3: Algebraic expressions
WelcomeAlgebraic ExpressionsSimplified sumsGreatest common factorFactoringExercisesQuiz ReviewExample 18:Factor60x2y5+ 108x3y4+ 120x8ycompletely.We need to find the GCF of the three monomials. To do this, first factor and find theGCF of their coefficients:60 = 22·3·5108 = 223350120 = 23·3·5The GCF of the coefficients is the product of lowest powers:22·31= 12.
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