## Expand: (x^2-6x+9)(x^2+8x+13)

Quadratic equations and inequalities, variation equations, function notation, systems of equations, etc.

### Expand: (x^2-6x+9)(x^2+8x+13)

How should I start expanding two quadratics? Do I factor and then do it the long way?

(x^2-6x+9)(x^2+8x+13)
bird

Posts: 1
Joined: Tue Feb 03, 2009 12:47 am

bird wrote:How should I start expanding two quadratics? Do I factor and then do it the long way?

Factoring is going in the wrong direction. You're not wanting to take this apart; you're wanting to multiply it all together.

bird wrote:(x^2-6x+9)(x^2+8x+13)

Once you have this many terms, vertical multiplication is probably the way to go. It's just so much simpler, usually...

So, just like you use vertical multiplication for multiplying three-digit and larger numbers, use a vertical set-up here:

`start like this:     x^2 - 6x +  9     x^2 + 8x + 13------------------`

Then you'll multiply, starting with the right-most term of the lower row, just like in regular (numerical) multiplication:

`first multiplication:     x^2 -  6x +   9     x^2 +  8x +  13--------------------   13x^2 - 78x + 117`

Then you continue:

`second multiplication:         x^2 -  6x +   9         x^2 +  8x +  13------------------------       13x^2 - 78x + 1178x^3 - 56x^2 + 72x`

stapel_eliz

Posts: 1803
Joined: Mon Dec 08, 2008 4:22 pm

### Re: Expand: (x^2-6x+9)(x^2+8x+13)

When you first start two expand two polynomials expressions(which is a binomial expression just with more than two terms), you must first multiply each term in the first expression by each term of the opposite expression. So let's start:
Step 1
Multiply the first x^2 to each term in the opposing expression.
You get x^4+8x^3+13x^2
Step 2
Multiply the -6x to each term of the opposing expression.
You get -6x^3-48x^2-78x
Step 3
Multiply the last term, which is 9, to each term of the opposing expression.
You get 9x^2+72x+117
Step 4
After you multiply all of the terms, you put them into one long expression making sure that your exponents go increasing to decresing left to right.
You get x^4+8x^3-6x^3+13x^2-48x^2+9x^2-78x+72x+117
Now that looks like a scary behemoth, but the whole point of putting them in order is so you can recognize like terms. So after you get this long expression, you combine your like terms to turn it into something you can handle.
Step 5
You get x^4+2^3-26x^2-6x+117
Your two polynomial expressions were turned into a behemoth expression and then turned back into a polynomial expression to the 4th degree.

Some helpful hints when expanding any expressions are: