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:  12 = 2 \cdot 2 \cdot 3
  • For another example, we can factor the number 30 into: 30 = 2 \cdot 2 \cdot 3 \cdot 5

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.

algeb72

2 \cdot 3 = 6 \leftarrow 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.

algeb73

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: 12x^3 + 10x^2 - 6x

 

We’ll begin by factoring 12x^312x^3 = 2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot x

Next, 10x^2 factors into: 10x^2 = 2 \cdot 5 \cdot x \cdot x

Finally, 6x factors into: 6x = 2 \cdot 3 \cdot x

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.

algeb74

GCF  = 2 \cdot x = 2x

For the third step, we’ll write the GCF of 2x outside a set of parenthesis and divide each term of the original expression BY 2x to determine what goes inside the parenthesis.

12x^3 + 10x^2 - 6x = 2x\left( \frac{12x^3}{2x} + \frac{10x^2}{2x} - \frac{6x}{2x} \right)

= 2x(6x^2 + 5x - 3)

Therefore, the expression inside the parenthesis is: 6x^2 + 5x minus 3. This makes the final factored answer:  2x times the quantity: 6x^2 + 5x - 3. This answer is considered “factored” form since we have written the expression as a product of 2x and 6x^2 + 5x - 3

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.

2x times 6x^2 is equal to positive 12x^3

2x times 5x is equal to positive 10x^2

2x times -3 is equal to negative 6x

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: -45a^3 + 9a^2

 

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, -45a^3, is negative. For now, let’s disregard that negative and factor: 45a^3

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.

algeb75

So 45a^3 factors into: 45a^3 = 3 \cdot 3 \cdot 5 \cdot a \cdot a \cdot a

Next, 9a^2 factors into: 9a^2 = 3 \cdot 3 \cdot a \cdot a

Each of these factorizations has two 3’s and two “a’s” in common, which means the GCF is 9a^2. But since the leading term is negative, we’ll make the GCF:-9a^2. 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.

algeb76

GCF = 3 \cdot 3 \cdot a\cdot a = 9a^2

For the third step, we’ll write the GCF of -9a^2 outside a set of parenthesis and divide each term of the original expression by -9a^2 to determine what goes inside the parenthesis.

-45a^3 divided by -9a^2 simplifies to 5a

9a^2 divided by -9a^2 simplifies to -1

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: -9a^2 times the quantity:  5a – 1

-45a^3 + 9a^2 = -9a^2\left( \frac{-45a^3}{-9a^2} + \frac{9a^2}{-9a^2} \right)

-9a^2\left( \frac{-45a^3}{-9a^2} + \frac{9a^2}{-9a^2} \right) = -9a^2\left( \frac{-45a^3}{-9a^2} - \frac{9a^2}{9a^2} \right)

-9a^2\left( \frac{-45a^3}{-9a^2} - \frac{9a^2}{9a^2} \right) = -9a^2(5a - 1)

Let’s do a quick check by multiplying. To do so, we will distribute the -9a^2 and hope that the result is equal to our original expression.

-9a^2 times 5a is equal to -45a^3

-9a^2 times -1 is equal to 9a^2

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

You have seen 1 out of 15 free pages this month.
Get unlimited access, over 1000 practice questions for just $29.99. Enroll Now