[Instructor] In this video,
we will take a look at two examples of factor by grouping.
Factor by grouping is a factoring technique,
often used when we have an even number of terms,
typically four or six terms,
but remember the first step in any factoring problem
is always to factor out the greatest common factor.
If we don't factor out the greatest common factor,
the expression will not be factored completely.
So looking at our first expression,
we are not going to add these two like terms
because if we do, we would have three terms
and we could not used factor by grouping
when we have an odd number of terms.
So again, our first step is to factor out
the greatest common factor.
Looking at all four terms, the greatest common factor
is two y, we'll begin by factoring out the two y.
We factor out two y from 10 y cubed, we have five y squared.
We factor out two y from 20 y squared, we have 10 y,
giving us plus 10 y.
We factor two y from six y squared, we have three y,
giving us plus three y.
If we factor two y from 12 y, we're left with six,
giving us plus six.
Now that we factored out the greatest common factor, two y,
we will now apply the technique of factor by grouping
to the expression inside the brackets.
Step one is we divide or group the polynomial into halves
because we have four terms, the first half
will have two terms, the second half will have two terms
as well.
The next step is to factor out the greatest common factor
from the first half and the second half.
The greatest common factor between five y squared and 10 y
is five y, so we factor out five y
from just the first two terms.
We factor out five y from five y squared, we have a factor
of y, we factor out five y from 10 y, we have a factor
of two, giving us plus two.
And now we do the same to the second half,
the greatest common factor of three y plus six is three
because we factor our positive three, we write plus three.
We factor out three from three y, we're left with y.
We factor out three from six, we're left with two,
giving us plus two.
The last step is to factor out the common polynomial factor.
Look at these two products.
If we the technique of factor by grouping correctly.
The products will always have a common polynomial factor,
which in this case, we can see is y plus two.
Now we factor out y plus two from these two products.
Again, we still have the factor of two y,
and now if we factor out y plus two from both products,
we have the quantity y plus two
and we can see the remaining factor is five y plus three.
So this is now factored completely,
but we can now drop the brackets
and write the final factored form as two y
times the quantity y plus two
times the quantity five y plus three.
Let's look at our second example on the next slide.
Remember the first step is always to factor out
the greatest common factor.
The greatest common factor of all four terms is four x,
but because this first term is negative,
let's factor out negative four x
rather than positive four x.
If we factor our negative four x,
it's going to change the sign of each term.
If we factor our negative four x from negative 20 x cubed,
we have positive five x squared.
If we factor negative four x from negative 80 x squared,
we're left with positive 20 x, so we have plus 20 x.
If we factor negative four x from negative 12 x squared,
we have positive three x, so plus three x.
If we factor negative four x from negative 48 x,
we're left with positive 12, giving us plus 12.
Again we can check this by distributing the negative four x
if we need to.
And now we factor the polynomial inside the parenthesis
using the factor by grouping technique.
First step is divide the terms in half,
so we have two terms on the left, two terms on the right.
The next step is to factor out the greatest common factor
from the first half and the second half.
So we have negative four x and then greatest common factor
of five x squared plus 20 x is five x.
We factor five x from just these two terms,
which leaves us with x plus four,
which again we can check by distributing
and now we factor out the greatest common factor
from three x plus 12 which is three
because we factor out positive three, we have plus three.
We factor out three from three x plus 12,
we're left with x plus four.
Again notice how we do have a common binomial factor
of x plus four, which we now factor out of the two products.
So we have negative four x
times, we factor out the factor of x plus four,
which we can see leaves us
with the factor of five x plus three.
This is factored completely,
but now we can drop the brackets
and write the final factored form as negative four x
times the quantity x plus four
times the quantity five x plus three.
I hope you found this helpful.