Factoring a difficult equation

[french_cheese]

Hello, this problem has been bugging for quite some time and well no body cares about my explanation so here is the problem.

(x^2+2)^5/2 + 2x(x^2+2)^3/2 + x^2(square root(x^2+2))

I have identified x^2(square root(x^2+2)) to be x^2(x^2+2)^1/2 and we can factor out (x^2+2) but this is about as far as I have gone because no matter what I do, I can't seem to get the answer from the book.

The answer is

(square root(x^2+2))(x^2+x+1)^2

I would greatly appreciate it if someone could help me with this please, preferably by working the problem out from start to finish. Thankyou

 

[galactus]

\(\displaystyle (x^{2}+2)^{\frac{5}{2}}+2x(x^{2}+2)^{\frac{3}{2}}+x^{2}(x^{2}+2)^{\frac{1}{2}}\)

Factor out the smallest power of \(\displaystyle (x^{2}+2)\). That would be \(\displaystyle (x^{2}+2)^{\frac{1}{2}}\)

\(\displaystyle (x^{2}+2)^{\frac{1}{2}}\left[(x^{2}+2)^{2}+2x(x^{2}+2)+x^{2}\right]\)

To help out with the inside portion, let \(\displaystyle u=x^{2}+2\)

We get \(\displaystyle u^{2}+2xu+x^{2}\)

Group and factor: \(\displaystyle u^{2}+ux+ux+x^{2}\)

\(\displaystyle u(u+x)+x(u+x)=(u+x)(u+x)=(u+x)^{2}=(x^{2}+x+2)^{2}\)

\(\displaystyle =(x^{2}+2)^{\frac{1}{2}}(x^{2}+x+2)^{2}\)

The reason you can not get the book answer is because that book answer is incorrect as written.

 

[Mrspi]

Hello, this problem has been bugging for quite some time and well no body cares about my explanation

We certainly DO care about your explanation...if you show us what you've tried, we have a MUCH better chance of helping you.

 

[french_cheese]

Thank you galactus for your help, now that I see how you did it I feel kind of stupid.

At Mrspi, I explained where I came to be stuck with the problem, but next time I will be more thorough. If you were referring to the first sentence, well this problem was one our teacher told us to skip but I didn't pay attention and tried to do it anyway. When I asked her about it, she said don't worry about it as it wasn't an assigned problem, but it has been bugging for quite some time as I do not like to feel defeated by a math problem in pre calculus. I would rather feel defeated by an advanced math problem having to do with differential equations or something like that, but not by a simple problem I should know how to do.