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In this video I want to do a
bunch of examples of factoring

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a second degree polynomial,
which is

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often called a quadratic.

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Sometimes a quadratic
polynomial, or just a

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quadratic itself, or quadratic
expression, but all it means

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is a second degree polynomial.

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So something that's going to
have a variable raised to the

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second power.

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In this case, in all of the
examples we'll do, it'll be x.

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So let's say I have the
quadratic expression, x

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squared plus 10x, plus 9.

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And I want to factor it into the
product of two binomials.

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How do we do that?

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Well, let's just think about
what happens if we were to

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take x plus a, and multiply
that by x plus b.

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If we were to multiply these
two things, what happens?

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Well, we have a little bit
of experience doing this.

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This will be x times x, which is
x squared, plus x times b,

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which is bx, plus a times x,
plus a times b-- plus ab.

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Or if we want to add these two
in the middle right here,

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because they're both
coefficients of x.

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We could right this as x squared
plus-- I can write it

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as b plus a, or a plus
b, x, plus ab.

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So in general, if we assume that
this is the product of

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two binomials, we see that this
middle coefficient on the

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x term, or you could say the
first degree coefficient

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there, that's going to be
the sum of our a and b.

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And then the constant term is
going to be the product

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of our a and b.

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Notice, this would map
to this, and this

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would map to this.

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And, of course, this is the
same thing as this.

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So can we somehow pattern
match this to that?

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Is there some a and b where
a plus b is equal to 10?

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And a times b is equal to 9?

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Well, let's just think about
it a little bit.

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What are the factors of 9?

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What are the things that a
and b could be equal to?

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And we're assuming that
everything is an integer.

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And normally when we're
factoring, especially when

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we're beginning to factor,
we're dealing

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with integer numbers.

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So what are the factors of 9?

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They're 1, 3, and 9.

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So this could be a 3 and a 3,
or it could be a 1 and a 9.

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Now, if it's a 3 and a 3, then
you'll have 3 plus 3-- that

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doesn't equal 10.

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But if it's a 1 and a
9, 1 times 9 is 9.

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1 plus 9 is 10.

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So it does work.

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So a could be equal to 1, and
b could be equal to 9.

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So we could factor this
as being x plus 1,

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times x plus 9.

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And if you multiply these two
out, using the skills we

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developed in the last few
videos, you'll see that it is

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indeed x squared plus
10x, plus 9.

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So when you see something like
this, when the coefficient on

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the x squared term, or the
leading coefficient on this

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quadratic is a 1, you can just
say, all right, what two

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numbers add up to this
coefficient right here?

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And those same two numbers, when
you take their product,

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have to be equal to 9.

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And of course, this has to
be in standard form.

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Or if it's not in standard form,
you should put it in

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that form, so that you can
always say, OK, whatever's on

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the first degree coefficient,
my a and b

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have to add to that.

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Whatever's my constant term, my
a times b, the product has

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to be that.

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Let's do several
more examples.

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I think the more examples
we do the more

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sense this'll make.

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Let's say we had x squared
plus 10x, plus-- well, I

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already did 10x, let's do a
different number-- x squared

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plus 15x, plus 50.

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And we want to factor this.

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Well, same drill.

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We have an x squared term.

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We have a first degree term.

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This right here should be
the sum of two numbers.

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And then this term, the constant
term right here,

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should be the product
of two numbers.

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So we need to think of two
numbers that, when I multiply

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them I get 50, and when
I add them, I get 15.

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And this is going to be a bit of
an art that you're going to

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develop, but the more practice
you do, you're going to see

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that it'll start to
come naturally.

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So what could a and b be?

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Let's think about the
factors of 50.

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It could be 1 times 50.

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2 times 25.

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Let's see, 4 doesn't
go into 50.

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It could be 5 times 10.

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I think that's all of them.

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Let's try out these numbers,
and see if any of

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these add up to 15.

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So 1 plus 50 does not
add up to 15.

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2 plus 25 does not
add up to 15.

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But 5 plus 10 does
add up to 15.

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So this could be 5 plus 10, and
this could be 5 times 10.

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So if we were to factor this,
this would be equal to x plus

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5, times x plus 10.

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And multiply it out.

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I encourage you to multiply this
out, and see that this is

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indeed x squared plus
15x, plus 10.

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In fact, let's do it. x
times x, x squared. x

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times 10, plus 10x.

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5 times x, plus 5x.

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5 times 10, plus 50.

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Notice, the 5 times
10 gave us the 50.

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The 5x plus the 10x is giving
us the 15x in between.

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So it's x squared plus
15x, plus 50.

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Let's up the stakes a little
bit, introduce some negative

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signs in here.

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Let's say I had x squared
minus 11x, plus 24.

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Now, it's the exact
same principle.

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I need to think of two numbers,
that when I add them,

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need to be equal
to negative 11.

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a plus b need to be equal
to negative 11.

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And a times b need to
be equal to 24.

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Now, there's something for
you to think about.

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When I multiply both of these
numbers, I'm getting a

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positive number.

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I'm getting a 24.

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That means that both of these
need to be positive, or both

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of these need to be negative.

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That's the only way I'm going to
get a positive number here.

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Now, if when I add them, I get
a negative number, if these

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were positive, there's no way I
can add two positive numbers

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and get a negative number, so
the fact that their sum is

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negative, and the fact that
their product is positive,

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tells me that both a
and b are negative.

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a and b have to be negative.

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Remember, one can't be negative
and the other one

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can't be positive, because the
product would be negative.

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And they both can't be positive,
because when you add

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them it would get you
a positive number.

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So let's just think about
what a and b can be.

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So two negative numbers.

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So let's think about
the factors of 24.

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And we'll kind of have to think
of the negative factors.

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But let me see, it could be 1
times 24, 2 times 11, 3 times

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8, or 4 times 6.

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Now, which of these when I
multiply these-- well,

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obviously when I multiply
1 times 24, I get 24.

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When I get 2 times 11-- sorry,
this is 2 times 12.

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I get 24.

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So we know that all these,
the products are 24.

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But which two of these, which
two factors, when I add them,

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should I get 11?

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And then we could say,
let's take the

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negative of both of those.

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So when you look at these,
3 and 8 jump out.

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3 times 8 is equal to 24.

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3 plus 8 is equal to 11.

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But that doesn't quite
work out, right?

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Because we have a negative
11 here.

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But what if we did negative
3 and negative 8?

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Negative 3 times negative 8
is equal to positive 24.

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Negative 3 plus negative 8
is equal to negative 11.

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So negative 3 and
negative 8 work.

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So if we factor this, x squared
minus 11x, plus 24 is

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going to be equal to x minus
3, times x minus 8.

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Let's do another
one like that.

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Actually, let's mix it
up a little bit.

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Let's say I had x squared
plus 5x, minus 14.

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So here we have a different
situation.

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The product of my two numbers is
negative, right? a times b

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is equal to negative 14.

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My product is negative.

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That tells me that one of them
is positive, and one of them

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is negative.

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And when I add the two, a plus
b, it'd be equal to 5.

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So let's think about
the factors of 14.

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And what combinations of them,
when I add them, if one is

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positive and one is negative,
or I'm really kind of taking

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the difference of the
two, do I get 5?

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So if I take 1 and 14-- I'm just
going to try out things--

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1 and 14, negative 1 plus
14 is negative 13.

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Negative 1 plus 14 is 13.

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So let me write all of the
combinations that I could do.

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And eventually your brain
will just zone in on it.

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So you've got negative 1
plus 14 is equal to 13.

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And 1 plus negative 14 is
equal to negative 13.

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So those don't work.

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That doesn't equal 5.

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Now what about 2 and 7?

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If I do negative 2-- let me do
this in a different color-- if

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I do negative 2 plus 7,
that is equal to 5.

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We're done!

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That worked!

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I mean, we could have tried 2
plus negative 7, but that'd be

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equal to negative 5, so that
wouldn't have worked.

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But negative 2 plus 7 works.

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And negative 2 times
7 is negative 14.

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So there we have it.

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We know it's x minus
2, times x plus 7.

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That's pretty neat.

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Negative 2 times 7
is negative 14.

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Negative 2 plus 7
is positive 5.

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Let's do several more of these,
just to really get well

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honed this skill.

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So let's say we have x squared
minus x, minus 56.

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So the product of the two
numbers have to be minus 56,

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have to be negative 56.

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And their difference, because
one is going to be positive,

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and one is going to be
negative, right?

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Their difference has
to be negative 1.

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And the numbers that immediately
jump out in my

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brain-- and I don't know if they
jump out in your brain,

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we just learned this in
the times tables--

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56 is 8 times 7.

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I mean, there's other numbers.

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It's also 28 times 2.

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It's all sorts of things.

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But 8 times 7 really jumped
out into my brain, because

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they're very close
to each other.

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And we need numbers that are
very close to each other.

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And one of these has to be
positive, and one of these has

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to be negative.

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Now, the fact that when their
sum is negative, tells me that

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the larger of these two should
probably be negative.

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So if we take negative
8 times 7, that's

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equal to negative 56.

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And then if we take negative
8 plus 7, that is equal to

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negative 1, which is exactly the
coefficient right there.

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So when I factor this, this
is going to be x minus 8,

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times x plus 7.

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This is often one of the hardest
concepts people learn

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in algebra, because it
is a bit of an art.

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You have to look at all of the
factors here, play with the

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positive and negative signs,
see which of those factors

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when one is positive, one is
negative, add up to the

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coefficient on the x term.

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But as you do more and more
practice, you'll see that

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it'll become a bit
of second nature.

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Now let's step up the stakes
a little bit more.

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Let's say we had negative x
squared-- everything we've

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done so far had a positive
coefficient, a positive 1

243
00:12:49,040 --> 00:12:50,690
coefficient on the
x squared term.

244
00:12:50,690 --> 00:12:55,590
But let's say we had a
negative x squared

245
00:12:55,590 --> 00:12:59,440
minus 5x, plus 24.

246
00:12:59,440 --> 00:13:00,910
How do we do this?

247
00:13:00,910 --> 00:13:03,420
Well, the easiest way I can
think of doing it is factor

248
00:13:03,420 --> 00:13:05,670
out a negative 1, and then it
becomes just like the problems

249
00:13:05,670 --> 00:13:07,260
we've been doing before.

250
00:13:07,260 --> 00:13:11,660
So this is the same thing as
negative 1 times positive x

251
00:13:11,660 --> 00:13:15,990
squared, plus 5x, minus 24.

252
00:13:15,990 --> 00:13:16,300
Right?

253
00:13:16,300 --> 00:13:18,080
I just factored a
negative 1 out.

254
00:13:18,080 --> 00:13:20,180
You can multiply negative 1
times all of these, and you'll

255
00:13:20,180 --> 00:13:21,690
see it becomes this.

256
00:13:21,690 --> 00:13:23,555
Or you could factor the negative
1 out and divide all

257
00:13:23,555 --> 00:13:25,000
of these by negative 1.

258
00:13:25,000 --> 00:13:26,760
And you get that right there.

259
00:13:26,760 --> 00:13:29,360
Now, same game as before.

260
00:13:29,360 --> 00:13:33,510
I need two numbers, that when
I take their product I get

261
00:13:33,510 --> 00:13:34,690
negative 24.

262
00:13:34,690 --> 00:13:37,125
So one will be positive,
one will be negative.

263
00:13:37,125 --> 00:13:41,550


264
00:13:41,550 --> 00:13:43,770
When I take their sum,
it's going to be 5.

265
00:13:43,770 --> 00:13:48,850
So let's think about
24 is 1 and 24.

266
00:13:48,850 --> 00:13:55,750
Let's see, if this is negative 1
and 24, it'd be positive 23,

267
00:13:55,750 --> 00:13:57,540
if it was the other way around,
it'd be negative 23.

268
00:13:57,540 --> 00:13:58,500
Doesn't work.

269
00:13:58,500 --> 00:14:01,210
What about 2 and 12?

270
00:14:01,210 --> 00:14:04,530
Well, if this is negative--
remember, one of these has to

271
00:14:04,530 --> 00:14:05,040
be negative.

272
00:14:05,040 --> 00:14:07,570
If the 2 is negative, their
sum would be 10.

273
00:14:07,570 --> 00:14:09,750
If the 12 is negative, their
sum would be negative 10.

274
00:14:09,750 --> 00:14:11,300
Still doesn't work.

275
00:14:11,300 --> 00:14:13,290
3 and 8.

276
00:14:13,290 --> 00:14:16,850
If the 3 is negative,
their sum will be 5.

277
00:14:16,850 --> 00:14:17,910
So it works!

278
00:14:17,910 --> 00:14:24,620
So if we pick negative 3 and
8, negative 3 and 8 work.

279
00:14:24,620 --> 00:14:26,630
Because negative
3 plus 8 is 5.

280
00:14:26,630 --> 00:14:29,510
Negative 3 times 8
is negative 24.

281
00:14:29,510 --> 00:14:31,640
So this is going to be equal
to-- can't forget that

282
00:14:31,640 --> 00:14:35,230
negative 1 out front, and then
we factor the inside.

283
00:14:35,230 --> 00:14:42,990
Negative 1 times x minus
3, times x plus 8.

284
00:14:42,990 --> 00:14:44,800
And if you really wanted to,
you could multiply the

285
00:14:44,800 --> 00:14:46,450
negative 1 times this,
you would get 3

286
00:14:46,450 --> 00:14:47,560
minus x if you did.

287
00:14:47,560 --> 00:14:48,810
Or you don't have to.

288
00:14:48,810 --> 00:14:51,810


289
00:14:51,810 --> 00:14:53,220
Let's do one more of these.

290
00:14:53,220 --> 00:14:56,060
The more practice, the
better, I think.

291
00:14:56,060 --> 00:15:01,630
All right, let's say I had
negative x squared

292
00:15:01,630 --> 00:15:06,690
plus 18x, minus 72.

293
00:15:06,690 --> 00:15:09,320
So once again, I like to factor
out the negative 1.

294
00:15:09,320 --> 00:15:13,040
So this is equal to negative
1 times x squared,

295
00:15:13,040 --> 00:15:16,910
minus 18x, plus 72.

296
00:15:16,910 --> 00:15:19,650
Now we just have to think of
two numbers, that when I

297
00:15:19,650 --> 00:15:22,330
multiply them I get
positive 72.

298
00:15:22,330 --> 00:15:24,480
So they have to be
the same sign.

299
00:15:24,480 --> 00:15:26,930
And that makes it easier in our
head, at least in my head.

300
00:15:26,930 --> 00:15:29,260
When I multiply them,
I get positive 72.

301
00:15:29,260 --> 00:15:32,040
When I add them, I
get negative 18.

302
00:15:32,040 --> 00:15:34,410
So they're the same sign, and
their sum is a negative

303
00:15:34,410 --> 00:15:36,470
number, they both must
be negative.

304
00:15:36,470 --> 00:15:41,470


305
00:15:41,470 --> 00:15:43,790
And we could go through all
of the factors of 72.

306
00:15:43,790 --> 00:15:48,510
But the one that springs up,
maybe you think of 8 times 9,

307
00:15:48,510 --> 00:15:53,270
but 8 times 9, or negative 8
minus 9, or negative 8 plus

308
00:15:53,270 --> 00:15:55,210
negative 9, doesn't work.

309
00:15:55,210 --> 00:15:58,060
That turns into 17.

310
00:15:58,060 --> 00:15:58,860
That was close.

311
00:15:58,860 --> 00:15:59,560
Let me show you that.

312
00:15:59,560 --> 00:16:04,400
Negative 9 plus negative 8, that
is equal to negative 17.

313
00:16:04,400 --> 00:16:06,140
Close, but no cigar.

314
00:16:06,140 --> 00:16:06,990
So what other ones are there?

315
00:16:06,990 --> 00:16:08,005
We have 6 and 12.

316
00:16:08,005 --> 00:16:09,705
That actually seems
pretty good.

317
00:16:09,705 --> 00:16:13,710
If we have negative 6 plus
negative 12, that is equal to

318
00:16:13,710 --> 00:16:15,360
negative 18.

319
00:16:15,360 --> 00:16:16,590
Notice, it's a bit of an art.

320
00:16:16,590 --> 00:16:18,810
You have to try the different
factors here.

321
00:16:18,810 --> 00:16:22,360
So this will become negative
1-- don't want to forget

322
00:16:22,360 --> 00:16:29,440
that-- times x minus 6,
times x minus 12.

323
00:16:29,440 --> 00:16:29,866