Video Transcript
Factor Polynomials with a Common
Factor
So if we’re given six 𝑥 plus
fifteen and we’re asked to factor this, we’re looking for a “greatest common factor”
or “GCF.” So we’re looking for a greatest
common factor that goes into both six 𝑥 and fifteen. Well we know that they’re both in
the three times table; so three goes into them and there isn’t a bigger number. So we’ll write both of these as
with a factor of three. So I could say six 𝑥 is three
multiplied by two 𝑥 and fifteen is three multiplied by five.
Now using the distributive
property, we’re going to take that three out of each term and put it on the outside
of the parentheses. And then on the inside, we’ll have
what’s left over from each term. We’ll have first of all two 𝑥 and
then five. So now we have fully factored the
polynomial six 𝑥 plus fifteen by finding a greatest common factor of both terms and
putting that outside the parentheses.
Now we must factor the expression
seven 𝑥 cubed plus fourteen 𝑥 squared minus thirty-five 𝑥. So again what we’re looking for is
a greatest common factor of each of those terms. So looking at the numbers first of
all, we can see that each of those numbers are in the seven times table. So the greatest common factor is
seven first of all. And then looking at the variables,
we can see that each of them have an 𝑥; so the greatest common factor of all of
those terms will be seven 𝑥.
Now if we do exactly the same as
we did in the previous example and will write each of those terms with a
multiplication of seven 𝑥, you can see that the first term will be seven 𝑥
multiplied by 𝑥 and by 𝑥 again — so seven 𝑥 multiplied by 𝑥 squared. The next term, we know that seven
multiplied by two is fourteen. So we’ve got seven 𝑥 multiplied
by two first of all. And then focusing on the
variables, we know that 𝑥 multiplied by 𝑥 is 𝑥 squared. So it’ll be seven 𝑥 multiplied by
two 𝑥.
And now our last term, so let’s
look at it as a whole term of negative thirty-five 𝑥. And we know that seven multiplied
by negative five is negative thirty-five. So now we have seven 𝑥 multiplied
by negative five. And now again using the
distributive law, we’re going to look at- each of those terms have a seven 𝑥
inside; so we’re going to take seven 𝑥 and put it on the outside of the
parentheses. And then on the inside, we’ve got
each of those terms in blue. So first of all, we have 𝑥
squared plus two 𝑥 minus five. And now we have it; we have fully
factored this polynomial.
So just as a recap of what we
did. We looked at each of the terms and
we tried to find the greatest common factor. So we looked at the numbers first
and we found that seven was common in each of the terms. And then looking for the
variables, we could see that 𝑥 was common in each of the terms. So the greatest common factor was
seven 𝑥. Then we looked at each of the
terms individually and said seven 𝑥 multiplied by what is that previous term. And then once we’ve done that, we
used the distributive law to take seven 𝑥 on the outside of the parentheses,
leaving us with the terms left over.
Now let’s look at one with not
just 𝑥, but also 𝑦.
So we have to factor twelve 𝑥
squared 𝑦 to the power of five minus thirty 𝑥 to the power of four 𝑦 squared. So we’re looking first of all for
the greatest common factor in numbers. So we can see that two goes into
both; well, that’s quite small. So maybe there’s a larger one, but
we know three goes in. Then if two and three goes in,
then that means that six must go in. And there isn’t a bigger number
than that. So we’ve got the greatest common
factor as six.
Now looking at the variables,
let’s first focus on 𝑥. So we can see that the smallest
power of 𝑥 is 𝑥 squared, so 𝑥 squared goes into both of them. And the smallest power of 𝑦 is 𝑦
squared, so 𝑦 squared goes into both. So our greatest common factor is a
little bit more complicated this time; our greatest common factor is six 𝑥 squared
𝑦 squared, so we gonna do exactly as we did in the previous two examples. And we’re gonna take each of those
terms and write them as multiplications with the greatest common factor. So as always we do the numbers
first, so we can say that six multiplied by two is twelve.
Then looking at the 𝑥 squared
term, we can see that 𝑥 squared multiplied by one is 𝑥 squared, so we don’t need
to do anything with that. But we know that 𝑦 to the power
of five is 𝑦 multiplied by 𝑦 multiplied by 𝑦 multiplied by 𝑦 multiplied by 𝑦. And we know that 𝑦 squared is
just 𝑦 multiplied by 𝑦, so we can see that we’ve got three left over 𝑦s. So we’ll need to multiply our term
by 𝑦 to the power of three or 𝑦 cubed. Now so we don’t get confused,
we’re going to take this next term as all of negative thirty 𝑥 to the power of four
𝑦 squared. So six multiplied by negative five
is negative thirty.
And then looking at the 𝑥 powers,
we can see we’ve got 𝑥 squared in our greatest common factor. And we need to get two 𝑥 to the
power of four. So we know that 𝑥 to the power of
four is 𝑥 multiplied by 𝑥 multiplied by 𝑥 multiplied by 𝑥. And we know that 𝑥 squared is
just 𝑥 multiplied by 𝑥. So we can see that we’ve got two
left over to get to our term, so that will be negative five 𝑥 squared.
Now we’ve done all we need to
do. So we’re going to use the
distributive law again by taking our greatest common factor and putting that on the
outside of the parentheses. And then what we got left over is
two 𝑦 cubed minus five 𝑥 squared and there we have it. So just to recap what we did, we
went to a polynomial and we looked for the greatest common factor first by looking
at the constants. So we said what of twelve and
negative thirty; what’s their greatest common factor? And we can see that that was
six. Then we looked at the variables
and found the lowest power of each one; that was 𝑥 squared and 𝑦 squared. We then looked at each term
individually and said what do I have to multiply this greatest common factor by to
get the term above. So we found that that we’ll use
the distributive law, putting that on the outside of the parentheses and leaving us
with two 𝑦 cubed minus five 𝑥 squared.
