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Factoring out a constant

Polynomials can often be simplified by factoring.  Factoring means breaking a polynomial down into its "factors": variables or terms that are multiplied together to create the polynomial.

The most basic form of factoring is to "factor out a variable," which means to divide each term in the polynomial by a common factor. You can think of this as a sort of "reverse distributive property" process.  Rather than multiplying each term by something, you're dividing each term by something.

The main rule you want to remember is that, in order to factor out a variable, you need to be able to divide every term in the expression by the same factor. 

So if you have $x^2+5x$, you can factor out x because both terms, $x^2$ and $5x$ can be divided by $x$.  

But, if you have $x^2+6x+3$ you can't factor by anything.  $x^2$ and $6x$ can be divided by $x$ and $6x$ and $3$ can be divided by $3$, but nothing goes into all of the terms. 

Once you find a term that you can factor by, you divide each term by the same factor and put the factor outside the expressin (which will be in parentheses now).  Look at your final product and you'll see that what you're doing is essentially the reverse of distributive property!


Simplify: $2x^2 + 4x+12$

What can you divide each term by? $2$!


You end up with a distributive property expression (for more, see lesson Distributive Property).  To test if you factored correctly, distribute and see if you get your original expression:

$2(x^2+2x+6)=(2\times x^2)+(2\times 2x)+(2\times6)=2x^2+4x+12$

Yes!  The process worked!

You will often use factoring out a constant in conjunction with other types of factoring, but it's always a good first thing to think about when you want to simplify a polynomial expression.

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