Download Math 25 Activity 9: GCF and Factoring Introduction - Math

Math 25 Activity 9: GCF and Factoring Introduction
This week’s activity is an opening for next week’s. We want to be able to understand factoring and
be able to factor easily, quickly, and accurately. This week we will discuss common factors and the
basics of factoring and next week we will discuss more difficult factoring problems.
Warm up with this application problem with your group.
1. Imagine you have to make gift baskets as raffle prizes at a talent show and you want all the
baskets to have the same contents. This being the last show, you want to use up your entire prize
inventory from previous years. What is the maximum number of baskets can you make if you have
24 chocolate bars, 40 movie passes, and 16 stuffed toys in the prize inventory?
The answer should be that the most you can make is eight baskets that are exactly the same and
that use up all your supplies. The idea is that you want to divide 24, 40, and 16 by the same number
and you want that number as large as possible. The mathematical concept behind this question is
the greatest common factor (GCF).
Factoring is a process that takes a complex algebraic expression and rewrites it as a product of two
simpler expressions. When we say something is “factored completely” we mean that we have a
product of simple expressions that cannot be simplified any more.
The most basic example of factoring is that of finding a prime factorization of a number. All
numbers have a unique prime factorization, meaning we can write a number as a product of primes
and there is only one end result. Here is an example of the prime factorization of 12 and 18.
Notice how 2 and 3 are prime numbers so we cannot make this expression any simpler.
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To be able to factor more complex expressions we need to review the concept of the greatest
common factor between two numbers or algebraic expressions. The greatest common factor (GCF)
of two numbers is unique. It is the largest number of all the possible numbers that evenly divides
both the numbers.
2. Find the greatest common factor of 48 and 56.
3. Did you use a method to find the GCF that would help you find the GCF of 40,500 and 15,750?
Some methods are straight forward, but can become tedious when numbers get large. We can think
of primes as the building blocks of all numbers. If we want to know all the common factors, then we
can simply look at all the common primes. The greatest common factor will be the product of all the
common primes. Here is an example of finding the GCF of 12 and 18.
Let’s look at our bigger numbers for an example of using exponents for finding the GCF. Find the
GCF of 40,500 and 15,750 by first finding the prime factorizations and writing them as a product of
primes with exponents.
40,500  2 2  34  53
15,750  2  32  53  7
The primes they share are 2, 3, and 5. Since they do not share 7, then we will not include that
number in the GCF. The smallest exponent on 2 is that it only appears once in the number 15,750.
The smallest exponent on the 3 is two, which means we need two of them. The exponent on the 5
is three in both numbers, so we need three of them. Combined we have 2  32  53  2,250 which
means the GCF of 40,500 and 15,750 is 2,250.
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We treat different variables like individual prime numbers, which means we can find the GCF of
algebraic expressions.
4. Find the GCF of
and
.
So far we have only discusses monomials, meaning one term, but factoring also applies to
polynomials that have more terms. Let’s look at factoring binomials first.
Recall the distributive property: a(b + c) = a  b + a  c . Usually students think of this property and
use it when they use distribution.
5. Simplify 5(x+y)
We call it distribution when we multiply the 5 by the x and the 5 by the y which gives you 5x+5y.
Factoring is using the other part of the property. If you are given a polynomial in which all the terms
have a factor in common, then you can divide each term by the factor and put it in front of the
parentheses, indicating multiplication to all terms. When directions tell you to factor, then you need
to factor out the greatest common factor from all the terms.
Example: Factor 7a+7c
7 is the GCF of 7a and 7c, which means I can divide both by 7. I am left with a+c so the answer is
7(a+c).
Example: Factor 8nm-16nx+48ny
8n is the GCF of 8nm, 16nx, and 48ny, which means I can divide all by 8n. I am left with m-2x+6y so
the answer is 8n(m-2x+6y).
Practice the following factoring problems, factoring out the GCF is the important first step that
needs to be checked in every case of factoring, which we will continue in the next activity.
6. Factor 3x+9y
7. Factor 16xz-4xp
8. Factor
9. Factor
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