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Algebraic Equations and Inequalities

# GED Mathematical Reasoning: Factoring Expressions I – GCF

Factoring is the process of writing a number or expression as a product. Put another way, factoring “undoes” multiplication. For a simple example, you might factor the number 6 into the product of 3 times 2, where the numbers 3 and 2 are called “factors.”

It is common to factor a number or expression into PRIME factors. A “prime” factor is divisible only by itself and one.

• Examples of PRIME numbers are: 2,3, 5, 7, 11, 13, 17, 19, 23
• For example, we can factor the number 12 into:
• For another example, we can factor the number 30 into:

Do you see that all these factors are prime? Also notice that the numbers 12 and 30 have a couple factors in common:  2 and 3

Multiplying these common factors of 2 and 3 together gives us what we call the “greatest common factor”, or G-C-F, of 12 and 30.

Greatest Common Factor (GCF)

Before we move on, let me share with you a quick and simple way to factor a number into prime factors. It is called the “factor tree” method.

• Consider the number 30. Choose two numbers that multiply to be 30. I’ll go with:  2 and 15
• 2 is a prime number so we’ll leave that branch alone, but 15 can be factored further — into 3 times 5.
• 3 and 5 are both prime and so our factor tree is complete. The number appearing at the end of each branch makes up the prime factorization.

Factoring a polynomial with Greatest Common Factor

• The first step of factoring a polynomial by factoring out the GCF is to factor each term separately.
• Next, find the Greatest Common Factor (GCF) of all terms. If there is a leading negative sign, make the GCF negative. We’ll talk more about this in Example 2.
• For step 3, write the GCF on the outside of a set of parenthesis. Divide each term of the original expression by the GCF and simplify to determine what goes INSIDE the parenthesis.
• And finally, check by multiplying.

Example 1

Factor by factoring out the GCF:

We’ll begin by factoring

Next, factors into:

Finally, factors into:

Let’s pause here to note that factoring each term separately is not the same as factoring the whole expression. To factor the whole expression, we must write it as a product.

Now let’s move on to Step 2 and determine the GCF. Each of these factorizations has a 2 and an “x” in common, which means the GCF is 2x.

GCF  =

For the third step, we’ll write the GCF of  outside a set of parenthesis and divide each term of the original expression BY  to determine what goes inside the parenthesis.

Therefore, the expression inside the parenthesis is: . This makes the final factored answer:   times the quantity: . This answer is considered “factored” form since we have written the expression as a product of  and

Since factoring “undoes” multiplication, we can do a quick check by multiplying. To do so, we will distribute the 2x and hope that the result is equal to our original expression.

times is equal to positive

times  is equal to positive

times -3 is equal to negative

Since this is equal to the original expression, we can be confident that our answer is correct.

Example 2

Factor by factoring out the GCF:

We’ll follow the same four-step process to factor the GCF from the expression shown in Example 1, but this time notice that the leading term, , is negative. For now, let’s disregard that negative and factor:

Use a factor tree if it’s helpful to you. The number 45 can be broken down into the product of 5 times 9. 5 is prime, so we’re done with that branch, but 9 can be broken down further into 3 times 3.

So  factors into:

Next, factors into:

Each of these factorizations has two 3’s and two “a’s” in common, which means the GCF is . But since the leading term is negative, we’ll make the GCF:. We factor out a leading negative in this way, because it is widely agreed that we’d rather have a negative GCF than a leading negative term inside the parenthesis. You will come to understand this preference in your continued studies.

For the third step, we’ll write the GCF of outside a set of parenthesis and divide each term of the original expression by to determine what goes inside the parenthesis.

divided by simplifies to

divided by  simplifies to

Although it may be tempting to think that the 9 “a” squared terms cancel out completely, it’s important to recognize that the division results in a placeholder of “1.”

Therefore, the expression inside the parenthesis is:  5a – 1.

This makes the final factored answer:  times the quantity:  5a – 1

Let’s do a quick check by multiplying. To do so, we will distribute the  and hope that the result is equal to our original expression.

times is equal to

times is equal to

Since this is equal to the original expression, we can be confident that our answer is correct.

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