Davion Harding

2022-06-19

How to simplify equations with a constant times a polynomial to a power

I've started doing integration in class and I often come across parts like this towards the end of my integration:

$\frac{2(6{x}^{2}+6x+9{)}^{3/2}}{3}+C$

And I get stuck. How should I go about simplifying these? Do I distribute the two and then try to do something with the exponent, or am I not allowed to because the polynomial is being raised to a power?

All help is appreciated, I am really stuck because almost all of my problems end up in this form at some point or another.

I've started doing integration in class and I often come across parts like this towards the end of my integration:

$\frac{2(6{x}^{2}+6x+9{)}^{3/2}}{3}+C$

And I get stuck. How should I go about simplifying these? Do I distribute the two and then try to do something with the exponent, or am I not allowed to because the polynomial is being raised to a power?

All help is appreciated, I am really stuck because almost all of my problems end up in this form at some point or another.

enfujahl

Beginner2022-06-20Added 20 answers

This looks like it's in simplest form. You could check to see whether the inner polynomial factors nicely, but this one doesn't. All the coefficients of the polynomial are divisible by $3$, but it only makes things messier to pull out ${3}^{3/2}$

Incidentally, supposing the polynomial factored, you have to be careful when simplifying. For instance, if we make it ${x}^{2}$ instead of $6{x}^{2}$, then:

$\begin{array}{rl}({x}^{2}+6x+9{)}^{3/2}& ={((x+3{)}^{2})}^{3/2}\\ & =|x+3{|}^{3}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{(not}(x+3{)}^{3}\text{!)}\end{array}$

Incidentally, supposing the polynomial factored, you have to be careful when simplifying. For instance, if we make it ${x}^{2}$ instead of $6{x}^{2}$, then:

$\begin{array}{rl}({x}^{2}+6x+9{)}^{3/2}& ={((x+3{)}^{2})}^{3/2}\\ & =|x+3{|}^{3}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{(not}(x+3{)}^{3}\text{!)}\end{array}$