Professor Dave here, let’s learn how to factor polynomials.
Professor Dave here, let’s learn how to factor polynomials.

We just learned how to multiply binomials by the FOIL method, and understanding this
We just learned how to multiply binomials by the FOIL method, and understanding this

technique will allow us to express simple polynomials in the form AX squared plus BX
technique will allow us to express simple polynomials in the form AX squared plus BX

plus C equals zero.
plus C equals zero.

This is the standard form for a quadratic, or second degree polynomial, with A, B, and
This is the standard form for a quadratic, or second degree polynomial, with A, B, and

C being some integers.
C being some integers.

Remember that when we solve something like this, what we are asking is: what value of
Remember that when we solve something like this, what we are asking is: what value of

X will make this equation true?
X will make this equation true?

In other words, what number can I plug in for X such that this expression will indeed
In other words, what number can I plug in for X such that this expression will indeed

be equal to zero?
be equal to zero?

The first technique we will use to solve some simple trinomials will be the reverse of the
The first technique we will use to solve some simple trinomials will be the reverse of the

FOIL method we just learned, in that FOILing two binomials gives us a polynomial, whereas
FOIL method we just learned, in that FOILing two binomials gives us a polynomial, whereas

we will now take a polynomial and express it as the product of two binomials.
we will now take a polynomial and express it as the product of two binomials.

To do this we have to be a little clever and think backwards.
To do this we have to be a little clever and think backwards.

First we have to understand that when we multiply binomials, like X plus M times X plus N, we
First we have to understand that when we multiply binomials, like X plus M times X plus N, we

get X squared plus MX plus NX plus MN.
get X squared plus MX plus NX plus MN.

We can combine the X terms and get M plus N times X.
We can combine the X terms and get M plus N times X.

So what that means, is that if we have some polynomial in the form of AX squared plus
So what that means, is that if we have some polynomial in the form of AX squared plus

BX plus C, where A equals one, in order to find out which binomials could multiply together
BX plus C, where A equals one, in order to find out which binomials could multiply together

to get this polynomial, we need to find the two numbers, M and N, that go in the binomials,
to get this polynomial, we need to find the two numbers, M and N, that go in the binomials,

and they must be numbers that multiply together to give C, and add together to give B. Let’s
and they must be numbers that multiply together to give C, and add together to give B. Let’s

do an example and I’ll show you what I mean.
do an example and I’ll show you what I mean.

Take X squared plus seven X plus ten.
Take X squared plus seven X plus ten.

Because the first term is X squared, the binomials can both start with X, so we just have to
Because the first term is X squared, the binomials can both start with X, so we just have to

figure out which numbers go in these two spots.
figure out which numbers go in these two spots.

The two numbers must multiply to give ten and add to give seven, so let’s start by
The two numbers must multiply to give ten and add to give seven, so let’s start by

listing pairs of factors for ten.
listing pairs of factors for ten.

In this case there are very few options, plus or minus ten and one, and plus or minus five
In this case there are very few options, plus or minus ten and one, and plus or minus five

and two.
and two.

Remember, the product of two negatives is a positive, so negative factors are possible
Remember, the product of two negatives is a positive, so negative factors are possible

here.
here.

Now we ask ourselves, which of these pairs of factors can add or subtract in some way
Now we ask ourselves, which of these pairs of factors can add or subtract in some way

to give seven?
to give seven?

Ten and one won’t work, since we could get eleven or nine, either positive or negative,
Ten and one won’t work, since we could get eleven or nine, either positive or negative,

depending on the signs that go in the binomials, but we can’t get seven.
depending on the signs that go in the binomials, but we can’t get seven.

Five and two will work, however, since five plus two is seven.
Five and two will work, however, since five plus two is seven.

So that means we can put positive five here and positive two here, and we have successfully
So that means we can put positive five here and positive two here, and we have successfully

factored the polynomial.
factored the polynomial.

We can check our work by multiplying through.
We can check our work by multiplying through.

X squared plus two X plus five X plus ten, after combining like terms, does indeed give
X squared plus two X plus five X plus ten, after combining like terms, does indeed give

us the original expression.
us the original expression.

Notice that we add five and two to get the coefficient of seven X, and we multiply five
Notice that we add five and two to get the coefficient of seven X, and we multiply five

and two to get ten, which is precisely what we were doing in reverse by selecting five
and two to get ten, which is precisely what we were doing in reverse by selecting five

and two.
and two.

Let’s try another.
Let’s try another.

X squared minus eight X minus twenty.
X squared minus eight X minus twenty.

Just like before, we always start by listing the pairs of possible factors for the C term.
Just like before, we always start by listing the pairs of possible factors for the C term.

Things get a little trickier when the last term is negative, since we have one and negative
Things get a little trickier when the last term is negative, since we have one and negative

twenty, but also negative one and positive twenty.
twenty, but also negative one and positive twenty.

Then there are both combinations of two and ten, as well as five and four, so we have
Then there are both combinations of two and ten, as well as five and four, so we have

six pairs of factors that might work.
six pairs of factors that might work.

But once again, though any of these pairs will multiply to give negative twenty, only
But once again, though any of these pairs will multiply to give negative twenty, only

one of them will add to give negative eight.
one of them will add to give negative eight.

We go through the options, trying the math in our head.
We go through the options, trying the math in our head.

Five and four is no good, nor is one and twenty, but with ten and two, we could have negative
Five and four is no good, nor is one and twenty, but with ten and two, we could have negative

ten and positive two, and that does add to negative eight.
ten and positive two, and that does add to negative eight.

So we can write X minus ten times X plus two, and that’s the answer, which we can then
So we can write X minus ten times X plus two, and that’s the answer, which we can then

FOIL if we want to check the answer.
FOIL if we want to check the answer.

Let’s do one more that is a little trickier.
Let’s do one more that is a little trickier.

Three X squared plus five X minus two.
Three X squared plus five X minus two.

Now that there is a coefficient of three by the X squared, that means one of the binomials
Now that there is a coefficient of three by the X squared, that means one of the binomials

has to start with three X, and the other with X, because that’s the only way to get three
has to start with three X, and the other with X, because that’s the only way to get three

X squared for the first term.
X squared for the first term.

Now that also means that it is not arbitrary which other number goes where, because one
Now that also means that it is not arbitrary which other number goes where, because one

will multiply by three X and the other by X.
will multiply by three X and the other by X.

Nevertheless, let’s get our factors and see what happens.
Nevertheless, let’s get our factors and see what happens.

We have negative two, so luckily there are only two options, it’s either positive one
We have negative two, so luckily there are only two options, it’s either positive one

and negative two, or negative one and positive two.
and negative two, or negative one and positive two.

So we have to ask ourselves, which pair of factors will allow us to multiply one factor
So we have to ask ourselves, which pair of factors will allow us to multiply one factor

by three X and the other by X, and add those together to get five X.
by three X and the other by X, and add those together to get five X.

First we try one and negative two.
First we try one and negative two.

That could give us three X minus two X, but that’s no good.
That could give us three X minus two X, but that’s no good.

X minus six X also doesn’t work.
X minus six X also doesn’t work.

How about negative one and positive two.
How about negative one and positive two.

We quickly see that if we multiply the two by three X and the negative one by X, we get
We quickly see that if we multiply the two by three X and the negative one by X, we get

six X minus X, or five X.
six X minus X, or five X.

So we place the negative one and the two into these binomials making sure to put them in
So we place the negative one and the two into these binomials making sure to put them in

the correct spots.
the correct spots.

We need the two to multiply the three X to get six X, so we have to place it in the other
We need the two to multiply the three X to get six X, so we have to place it in the other

binomial so that they do indeed multiply.
binomial so that they do indeed multiply.

We do not put it in the same binomial as three X.
We do not put it in the same binomial as three X.

And there we have it, three X minus one times X plus two, which once again, we can FOIL
And there we have it, three X minus one times X plus two, which once again, we can FOIL

to double check our answer.
to double check our answer.

At this point, you may be wondering, why are we doing this?
At this point, you may be wondering, why are we doing this?

How does this help us solve polynomials?
How does this help us solve polynomials?

Well the answer is simple.
Well the answer is simple.

If any of these polynomials had been set equal to zero, then the binomials must multiply
If any of these polynomials had been set equal to zero, then the binomials must multiply

to give zero.
to give zero.

And we know that anything times zero is zero, so finding a value for X that will make either
And we know that anything times zero is zero, so finding a value for X that will make either

of these binomials equal to zero will necessarily satisfy the equation.
of these binomials equal to zero will necessarily satisfy the equation.

For example, with our first attempt, we factored this expression to get X plus two times X
For example, with our first attempt, we factored this expression to get X plus two times X

plus five.
plus five.

If that had been set equal to zero, we can now look at the first term.
If that had been set equal to zero, we can now look at the first term.

How do we get X plus two to be equal to zero?
How do we get X plus two to be equal to zero?

Well that’s easy, if X plus two equals zero, then X must be negative two, because negative
Well that’s easy, if X plus two equals zero, then X must be negative two, because negative

two plus two is zero.
two plus two is zero.

That means that if X is negative two, this first term is zero, and then it doesn’t
That means that if X is negative two, this first term is zero, and then it doesn’t

matter what this other term is, because zero times anything is zero.
matter what this other term is, because zero times anything is zero.

Likewise, if X is negative five, this term is zero, and the whole thing is zero.
Likewise, if X is negative five, this term is zero, and the whole thing is zero.

That means that negative two and negative five are the two solutions to this quadratic,
That means that negative two and negative five are the two solutions to this quadratic,

which we found simply by factoring the expression.
which we found simply by factoring the expression.

The second one we did gave us X minus ten times X plus two.
The second one we did gave us X minus ten times X plus two.

By similar reasoning, either positive ten or negative two can be plugged in for X to
By similar reasoning, either positive ten or negative two can be plugged in for X to

make this equation valid, so those are the two solutions.
make this equation valid, so those are the two solutions.

For the last one, we ended up with three X minus one times X plus two.
For the last one, we ended up with three X minus one times X plus two.

Of course negative two works, but for this other one we just need one extra step.
Of course negative two works, but for this other one we just need one extra step.

Let’s make a separate equation, where three X minus one equals zero.
Let’s make a separate equation, where three X minus one equals zero.

Add one to both sides, divide by three, and we get X equals one third, so those are the
Add one to both sides, divide by three, and we get X equals one third, so those are the

two solutions in that case.
two solutions in that case.

For any of these examples, we can always check our answers by plugging them into the original
For any of these examples, we can always check our answers by plugging them into the original

polynomial to verify that it does indeed equal zero, which is great for peace of mind on
polynomial to verify that it does indeed equal zero, which is great for peace of mind on

a test.
a test.

And with that, we have learned a simple technique to solve polynomials.
And with that, we have learned a simple technique to solve polynomials.

Not all polynomials can be solved by this method, as we will later see, but lots of
Not all polynomials can be solved by this method, as we will later see, but lots of

them will, so let’s check comprehension.
them will, so let’s check comprehension.