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Step 2Factoring the GCF from an expression
And no, I don't mean factoring the expression of your boss as you tell him you accidentally flooded the break room with coffee.
Algebraic expressions consist of numbers, which are called coefficients, and variables, which can be raised to a power. In the expression x^2 + 6x + 8, 1 is the coefficient of x^2, the variable. (If you do not see a coefficient before a variable, it's a 1, because x^2 is multiplied by 1.) Likewise, 6 is a coefficient of x^1. (A lone variable is raised to a power of one.) 8 is called a constant - it is not multiplied by a variable. (You can visualize it being multiplied by x^0, and any number raised to the 0th power is equal to 1).
To factor an expression, you have to start by factoring out the GCF, or Greatest Common Factor. List the factors of each component of the equation. Here we are interested in finding the natural number factors.
The expression x^2 + 6x + 8 would have factors that look like this:
x^2: 1
6x: 1, 2, 3, 6
8: 1, 2, 4, 8
If you look at the three lists, there is only one thing that they all share in common, the number one. This means there is no coefficient greater than one to factor out.
Then you look at the exponents' powers. 2, 1, and 0. If you see a zero, the expression cannot be factored by a variable.
This expression is ready for the next step.
Here is an example that does have a GCF that needs to be factored out: 2x^3 + 18x^2 + 10x. Factor each part:
2x^3: 1, 2
18x^2: 1, 2, 3, 6, 9, 18
10x: 1, 2, 5, 10
Here we can see that the parts have 1 and 2 in common. We find the largest number, 2.
Then we look at the powers of exponents: 3, 2, and 1. Find the smallest number that isn't 0, in this case the number one. That means x^1, or simply x, can be divided into the expression.
Multiply the number and variable together to get 2x. Then divide each part of the expression by 2x.
2x^3 / 2x = x^2
18x^2 / 2x = 9x
10x / 2x = 5
The expression with the GCF factored out is 2x (x^2 + 9x + 5). Note that you must put the factored expression in parentheses and write the GCF next to it.
Algebraic expressions consist of numbers, which are called coefficients, and variables, which can be raised to a power. In the expression x^2 + 6x + 8, 1 is the coefficient of x^2, the variable. (If you do not see a coefficient before a variable, it's a 1, because x^2 is multiplied by 1.) Likewise, 6 is a coefficient of x^1. (A lone variable is raised to a power of one.) 8 is called a constant - it is not multiplied by a variable. (You can visualize it being multiplied by x^0, and any number raised to the 0th power is equal to 1).
To factor an expression, you have to start by factoring out the GCF, or Greatest Common Factor. List the factors of each component of the equation. Here we are interested in finding the natural number factors.
The expression x^2 + 6x + 8 would have factors that look like this:
x^2: 1
6x: 1, 2, 3, 6
8: 1, 2, 4, 8
If you look at the three lists, there is only one thing that they all share in common, the number one. This means there is no coefficient greater than one to factor out.
Then you look at the exponents' powers. 2, 1, and 0. If you see a zero, the expression cannot be factored by a variable.
This expression is ready for the next step.
Here is an example that does have a GCF that needs to be factored out: 2x^3 + 18x^2 + 10x. Factor each part:
2x^3: 1, 2
18x^2: 1, 2, 3, 6, 9, 18
10x: 1, 2, 5, 10
Here we can see that the parts have 1 and 2 in common. We find the largest number, 2.
Then we look at the powers of exponents: 3, 2, and 1. Find the smallest number that isn't 0, in this case the number one. That means x^1, or simply x, can be divided into the expression.
Multiply the number and variable together to get 2x. Then divide each part of the expression by 2x.
2x^3 / 2x = x^2
18x^2 / 2x = 9x
10x / 2x = 5
The expression with the GCF factored out is 2x (x^2 + 9x + 5). Note that you must put the factored expression in parentheses and write the GCF next to it.
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