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### Factoring Polynomials

```Date: 02/08/2003 at 13:27:40
From: Giovanni
Subject: Factoring polynomials

9y^4 + 12y^2 + 4

The first term in the polynomial is not raised to the 2nd degree, and
when I do change it so that it becomes a term to the 2nd degree, it's
hard for me to find a number whose sum is 12 and product is 12. I
don't know if I'm approaching this question in the right way.
```

```
Date: 02/08/2003 at 16:33:37
From: Doctor Kastner
Subject: Re: Factoring polynomials

Hi Giovanni -

When you are factoring polynomials, the first thing to think about is
the pattern. Let me give you an example. We know how to factor the
difference of squares:

y^2 - 4 = (y-2)(y+2)

and whenever we see the form

a^2 - b^2

we should factor it into

(a-b)(a+b)

In actuality, a and b may be complicated expressions, but that doesn't
matter. We just need to ask if it is something squared minus something
squared:

625y^4 - 16x^4

can be written as

(25y^2)^2 - (4x^2)^2

which factors into

(25y^2 - 4x^2)(25y^2+4x^2)

but notice that we have another difference of squares in the first
term

(25y^2 - 4x^2) = (5y)^2 - (2x)^2 = (5y-2x)(5y+2x)

so we have that

625y^4 - 16x^4 = (5y-2x)(5y+2x)(25y^2+4x^2)

So even though it started off looking very complicated, it was just a
couple of difference of squares.

Now back to your problem. We are used to the form ay^2+by+c, but
notice that ax^4+bx^2+c is pretty much the same thing. If we write it
like

a(x^2)^2 + b(x^2)^1 + c

we can let y = x^2, and now it looks like

ay^2 + by + c

And you could factor this, making sure that you put the x^2 back in at
the end. But we don't always need to do this explicit substitution.
Instead, just remember that the factored form will have an x^2 in it
instead of an x. Both of the signs are positive, so that means it will
be of the form

(__ x^2 + __)(__x^2+__)

if we can factor it at all. The numbers that can go into the first
slots are 3 and 3 or 9 and 1, while the numbers that can go into the
last slots are 4 and 1 or 2 and 2. Finally, we need the middle term to
be 12, and looking at the combinations we could pick for the slots,
one of them leaps out. Do you see which one it is?

more, or if you have any other questions.

- Doctor Kastner, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Factoring Expressions

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