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(male narrator) In this video,
we will look at how we can factor out
the greatest common factor out of an expression.
As we set this up,
consider the expression a times b, plus c.
In the past, we have multiplied this
by multiplying the a through, giving us ab plus ac.
As we factor, we're attempting to multiply in reverse--
going from the product to the factors
that were multiplied to give us that product.
As we do, the number out front turns out to be
the greatest common factor of our expression.
We will put the greatest common factor
in front of the parentheses
and divide each term by that greatest common factor.
What's left goes in the parentheses.
Notice in this example, both terms have an a in common
that goes outside of the parentheses.
When we factor out an a,
we're left with the b and the c-- or b plus c.
Let's take a look at some examples
that are a little more interesting,
where we find the GCF
and put it in front of the parentheses to factor.
In this problem,
we see that 9, 12, and 6 are all divisible by 3.
All the terms also have an x,
and the lowest exponent is squared.
3x squared is the greatest common factor,
which will go in front of the parentheses.
To find out what's left inside the parentheses,
we will divide each term by 3x squared
to see what we have left.
Reducing the first fraction-- 9 over 3--
gives us 3x squared, subtracting the exponents;
minus...12 divided by 3 is 4x, when we subtract the exponents;
plus...6 divided by 3 is 2, and the x squares divide out.
We have now factored this expression
to 3x squared, times 3x squared, minus 4x, plus 2.
Notice if we were to multiply the 3x squared
back through the parentheses by distributing,
we would get the original problem
that we started with.
Let's try another example
where we identify the greatest common factor
and pull it in front of the parentheses.
Looking at the numbers 21, 14, and 7,
we see they are all divisible by 7.
Also, every factor has an a.
Using the lowest exponent, we write a squared.
Every factor also has a b.
Using the lowest exponent, we write b squared.
This is what goes in front of the parentheses,
and then, we can divide each of these terms
by that common factor:
7a squared, b squared, all the way across.
As we do and reduce those fractions,
21 over 7 gives us 3; a squared; b cubed;
minus...14 over 7 gives us 2;
subtracting exponents leaves us with ab to the 5th;
plus...7 over 7 is 1;
and the a's and b's divide out completely.
Notice as we did this,
just because the term divided out completely
did not mean it disappeared.
When things divide out, we're left with a 1 behind.
This becomes our solution
when we factor out the greatest common factor.