Addition and Subtraction of suitable terms

[Knuckler]

Hello, I am going through Schaum's College algebra and am on the chapter of factoring. I am having problems with this technique when dealing with trinoials.

In one example 36x^4 + 15x^2 + 4
9x^2 is both added and subtracted, yet it is not explained how 9x^2 is derived. I thought you were suppose to add and subtract 2 * (product of the square root of the two terms)

In another example
u^8 - 14u^4 + 25
4u^4 is added and subtracted. Again I have no idea how 4 is derived as it is neither a factor of 25 or 14. Can anyone help please?

 

[tkhunny]

No, you need to CREATE 2xsquare root of the two items).

In the first, you are looking for 24x^2, but you've already 15, You need only 9 more.

The second it a hair trickie due to the sign of the middle term. You need -10 and you've already -14. Only 4 to go.

 

[galactus]

The general form of the given expression is \(\displaystyle a^{4}+a^{2}b^{2}+b^{4}\)

With your problem, this means \(\displaystyle a=\sqrt{6}x\) and \(\displaystyle b=\sqrt{2}\)

Because \(\displaystyle (\sqrt{6}x)^{4}=36x^{4}\) and \(\displaystyle (\sqrt{2})^{4}=4\)

But, the middle term \(\displaystyle a^{2}b^{2}=(\sqrt{6}x)^{2}(\sqrt{2})^{2}=12x^{2}\)

Adding \(\displaystyle a^{2}b^{2}=12x^{2}\) gives \(\displaystyle 24x^{2}\).

We now have a perfect square trinomial. \(\displaystyle a^{4}+2a^{2}b^{2}+b^{4}=(a^{2}+b^{2})^{2}\)

But, the problem has a \(\displaystyle 15x^{2}\), but we need a \(\displaystyle 24x^{2}\).

This is where the \(\displaystyle 9x^{2}\) comes in.

By adding the \(\displaystyle 9x^{2}\), it becomes \(\displaystyle 36x^{4}+24x^{2}+4=(6x^{2}+2)^{2}\)

Subtract off the \(\displaystyle 9x^{2}\) and we get \(\displaystyle (6x^{2}+2)^{2}-9x^{2}\)

Now, apply the difference of two squares factorization.

 

[Knuckler]

thank you all. So basically were just forcing the polynomial into a perfect square trinomial by getting 2 * (a * c) and replacing b with it, then taking the difference of the old term and new term and using that as to do the difference of 2 squares factorization?