Hi, I am currently on problem 1F-2 of the first problem set and am having trouble following the algebra. I have reached to this part:

10x^9(x^2 + 1)^10 + x^10[10(x2 + 1)^9(2x)].

I do not understand how they arrive at = 10(3x^2 + 1)x^9(x^2 + 1)^9.

Could someone please explain?

Question

Hi, I am currently on problem 1F-2 of the first problem set and am having trouble following the algebra. I have reached to this part:

10x^9(x^2 + 1)^10 + x^10[10(x2 + 1)^9(2x)].

I do not understand how they arrive at = 10(3x^2 + 1)x^9(x^2 + 1)^9.

Could someone please explain?

phi · Answer

$$ 10x^9(x^2 + 1)^{10} + x^{10}\cdot 10(x^2 + 1)^9(2x)  $$

factor out $ 10x^9 (x^2+1)^9 $ from each term:
$$ 10x^9 (x^2+1)^9\left( (x^2 + 1)+x\cdot 2x\right)  \
10x^9 (x^2+1)^9\left( x^2 + 1+2x^2\right) \
10x^9 (x^2+1)^9(3x^2+1) 
 $$