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(x + n), where m and n are integers. Now, think about what must be true.

Let's FOIL this right-hand side. We have x^2 + n * x + m * x + m * n or

x^2 + (n + m)x + m * n. So, in order for this equality to hold,

the coefficient of x on the left, namely -10, has to be equal to the coefficient

of x on the right, n + m, and then these constants,

21 and m * n also have to be equal That is n + m must be equal to -10 and m * n =

21. So let's start by looking at some factors

of 21. And whichever factors of 21 add up to

give us -10 will be the m and n we're looking for.

So, let's make a table here. So, m, n, and n+m.

All right. So, what are some factors of 21? We have

1 and 21. We have -1 and -21.

We have 3 and 7. And -3 and -7. When we add these values together, 1 + 21

= 22. This is not -10 that we're looking for.

-1 - 21 = -22, it's also not -10. 3 + 7 = 10,

so, it's not -10. But if we add these two factors together,

yes. These add up together to give us -10.

So, therefore, we find our m and n up here.

So our answer then, is x - 3 * x - 7. This method of factoring is sometimes

referred to as factoring by trial and error,

because we're trying different factors of 21 until we find the ones that add up to

-10. And let's verify to ourselves that indeed

we did get the right factorization here. So if we FOIL this, we get x^2 - 7x - 3x

+ 21, which is equal to x^2 - 10x + 21, which, yes, is our expression.

Now, sometimes you'll see these problems done as follows.

We still want to find these two numbers n and m that add up to -10 and multiply to

21, but then, this -10x will be rewritten as

follows. We still have the x^2, but then we have

these numbers down here, -3 and -7 that add up to -10.

So we'll rewrite -10x as -3x - 7x, and then we still have the + 21.

And then, we'll factor by grouping, so we'll group these first two terms

together and the last two terms together. We'll factor out an x from the first two

terms, which leaves us x - 3. And then, we'll factor out a -7 from the

last two terms, which also leaves us with an x - 3.

And then, we factor out this common binomial here, which leaves us with the

same answer of (x - 3) * (x - 7). Now, this is sometimes called the ac

method, because we want to look at a quadratic expression of the form ax^2 +

bx + c and find factors of a * c that add up to b,

and in our case, a was 1 and c was 21. Now, this ac method is more commonly used

when our a > 1. So let's see an example of that.

[SOUND] For example, lets factor 2x^2 - 7x + 3.

Lets compare it to the quadratic form ax^2 + bx + c.

So we see here, that our a = 2, our b = -7, and our c = 3.

Lets apply this ac method that was just described.

What we do is we multiply a and c together, so it's 2 * 3 which is 6.

And we want to look at factors of six that will add together to give us our b

or -7, and we find that the factors would be -1

and -6. Therefore, we rewrite this quadratic

expression as 2x^2 and then - 1 * x or just -x,

and then -6x + 3. And then, we factor by grouping and from

the first two terms, we can factor out an x and we're left with 2x - 1.

And from the last two terms, we can factor out a -3, and then, we're also

left with 2x - 1, which then we can factor out this common binomial,

which gives us our answer of (2x - 1) * (x - 3).

Now, how would we factor this by trial and error? In other words, we want to try

to write a quadratic expression 2x^2 - 7x + 3 as a product of two binomials.

But if we FOILed out these binomials, the first term would have to be 2x^2, so 2x

and x would work on our first slots and our last term after FOILing would have to

be 3. So what are our choices here for the

second slot? Well, we could put a +1 here and a +3 here because 1 * 3 = 3 or we put

-1 and -3, because -1 * -3 is also 3 or the other way around.

We could have -3 here and -1 here, as well as +3 here and +1 here.

Now, remember, that our outer and inner terms would have to combine to give us

this -7x. Therefore, this first choice here, this

+1, +3, as well as the last choice the +3, +1

can't work, because their outer and inner terms would combine to give us a positive

number, which means these middle two here would be our only possibilities.

Now, what happens if we FOIL these two binomials out? We'd get 2x^2 - 2x - 3x,

which is -5x and then + 3. So our middle term is not right.

So we'll cross this off as well. But if we FOIL these two binomials out,

we would get to our original quadratic expression,

therefore, this is the factorization we found by the trial and error method,

which is the same answer that we found by the ac method. Alright.

And this is how we factor a quadratic expression.

Thank you and we'll see you next time. [SOUND]