A. Strategy 1: Prime Factorization
The following investigation explores greatest common factors of numbers and polynomials.
Investigation
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Complete parts 5, 6, 7, and 8 of Investigate Common Factors on p. 215 of Mathematics 10. |
The GCF of a group of monomials can be found by separately determining the GCF of the coefficients and the GCF of the variables. Multiplying these two GCFs gives the GCF for the group of monomials. Two possible strategies for determining the GCF of a group of monomials are prime factorization and listing the factors of each monomial.
Strategy 1: Prime Factorization
To use this strategy, show the prime factorization of each part of each expression and determine the common factors.
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The name of the strategy 'Prime Factorization' suggests identifying the prime factors of numbers. However, for the purposes of this unit, we are going to expand the meaning of prime factorization to include breaking powers into factors, each with an exponent
of 1. For example, we will call the prime factorization of |
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Example 1 |
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Determine the GCF of «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»36«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»48«/mn»«mi»x«/mi»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/math».
You do not need to separate the numbers from the variables to use this strategy, but it may be easier to follow if you do. An example where the monomials are kept intact is shown.
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and
is
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