Finding the Factor of an Equation

[college kid]

4-(x+y)²

How do you find a factor of a problem like this? I would post the work I have done so far, but I am honestly not sure how to start it. I have several problems to work like this so if someone could give me a start in the right direction, I would appreciate it. I already have the answer, but I need to know how to work problems like this. Thanks.

 

[JeffM]

4-(x+y)²

How do you find a factor of a problem like this? I would post the work I have done so far, but I am honestly not sure how to start it. I have several problems to work like this so if someone could give me a start in the right direction, I would appreciate it. I already have the answer, but I need to know how to work problems like this. Thanks.

This is not an equation. It is an expression. An equation has an = sign in it.

Just like your last post, this problem is seeing whether you remember a pattern.

Does the phrase difference of squares or difference of powers ring a bell.

\(\displaystyle a^2 - b^2 = (a - b)(a + b).\)

\(\displaystyle a^3 - b^3 = (a - b)(a^2 + ab + b^2).\)

\(\displaystyle a^4 - b^4 = (a - b)(a^3 + a^2b + ab^2 + b^3).\)

Multiply the expressions on the right out and simplify to validate. In general,

\(\displaystyle positive\ integer\ n \implies a^n - b^n = (a - b) * \left( \displaystyle \sum_{i=1}^na^{(n - i)}b^{(i - 1)}\right).\)

 

[college kid]

Thanks for all you help so far. I'm still a little confused though. I know that you have to break down this problem into a simpler form. According to the answer you end up with, it looks like you take the square root of everything ...

4-(x+y)²

2 + x + y

The square root of 4 is 2 and you would move the parentheses as you aren't squaring anything anymore. Sorry if this makes no sense, but I'm still not sure exactly what you are suppose to do. Thanks.

 

[mmm4444bot]

There is a common factoring pattern taught in elementary algebra courses; it's called "a difference of two squares".

Have you seen it before, perhaps? It looks like this:

a^2 - b^2 = (a + b)(a - b)


Your exercise fits this pattern because 4-(x+y)^2 is a difference of two squares.

2 is being squared

x+y is being squared

The latter square is being subtracted from the former square. That's a difference of two squares.


Try to fit the quantities 2 and x+y into the factoring pattern above for "a difference of two squares".

If you still do not understand how a difference of two squares factors, let us know, and we will find some lessons for you to study (with lots of examples). Cheers :cool: