Factoring Problem

[asdf]

I'm having trouble with this problem:

Factor: (4)(a^2)(c^2) - (a^2 - b^2 + c^2)^2

Thanks

[andi314]

==>
(4)(a^2)(c^2) - (a^2 - b^2 + c^2)^2 =

- (a + b + c)·(a + b - c)·(a - b - c)·(a - b + c)


hope i could help you
andi314

[asdf]

I have checked and the answer is correct, and I have accidentally found a way to find the factors, but it was through major guessing. Could anyone show how to factor this out fully?

[cheesehead]

There are online factorers. The one I frequently use is "Factoris" at http://wims.unice.fr/wims/wims.cgi It can factor an integer, a rational number, a polynomial or a rational function.

When given:

4(a^2)*(c^2)-(a^2-b^2+c^2)^2

Factoris returned:

Warning. You have entered an ambiguous formula, whose interpretation by wims may differ from what you meant. Please check the formula.

Please always use ``*'' for multiplication, and always enclose function arguments in parentheses. (For experts. If
you don't want wims to interprete your expression, start it with the string ``1-1+''.)

Factorization of the polynomial 4*(a^2)*(c^2)-(a^2-b^2+c^2)^2 into irreducible components over Z :

-a4 + (2c2 + 2b2) a2 + (-c4 + 2b2 c2 - b4)= -(c-b-a)(c-b+a)(c+b-a)(c+b+a)

Remark. Factorization of multivariate polynomials is only implemented over Z for the time being.

In the next-to-last line, Factoris superscripts the exponents, but that formatting is lost in my cut-and-pasting the text here. That is, what is meant is -a^4 + (2c^2 + 2b^2) a^2 + (-c^4 + 2b^2 c^2 - b^4) = -(c-b-a)(c-b+a)(c+b-a)(c+b+a)

Factoris doesn't use "*" for multiplication in its output even though it wants you to use it in input. :)

Of course, Factoris doesn't tell you how to factor an expression. But one might get clues from its answers.

[axn]

Originally Posted by asdf

I'm having trouble with this problem:

Factor: (4)(a^2)(c^2) - (a^2 - b^2 + c^2)^2

Thanks

Soln: (I am omitting the * for multiplication)

(4)(a^2)(c^2) - (a^2 - b^2 + c^2)^2

==> X^2 - Y^2, where X = 2ac and Y = (a^2 - b^2 + c^2)

==> (X+Y)(X-Y)

==> (2ac + a^2 - b^2 + c^2)(2ac - a^2 + b^2 - c^2)

==> [(a+c)^2 - b^2] [b^2 - (a-c)^2]

==>[(a+c+b)(a+c-b)] [(b+a-c)(b-a+c)]