Can't simplify this fraction: $ \frac{1+x^6}{1+x^2}$

Question

I've been having trouble simplifying this fraction : $$ \frac{1+x^6}{1+x^2} $$

Can anyone explain step by step on how to solve this?

Thank you.

Answer 1

$1+x^6=1+({x^2})^3$ which is equivalent to

$(1+x^2)(x^4+1-x^2)$ using $a^3+b^3=(a+b)(a^2+b^2-ab)$

Hence the expression becomes $$\frac{(1+x^2)(x^4+1-x^2)}{1+x^2} = (x^4+1-x^2).$$

Answer 2

For this rational function, or any other rational function, you can use polynomial long division to simplify it. In this case there is no remainder as $x^2 + 1$ is a factor of $x^6 + 1$, but if you didn't know that beforehand, you will find that out by applying long division.

Answer 3

For simple, you can let $x^2+1=a$. Then $1+x^6= a(a^2-a+1)$. Hence $\dfrac{x^6+1}{x^2+1}=a^2-a+1= x^4-x^2+1$.

Answer 4

You may also try it in Maple to find the desire insight for simplifying the fraction. This would be just a start point:

    [> factor(x^6+1);
                              (1+x^2)*(x^4-x^2+1)