Is there easy way to factor polynomials for calculus?

Question

factor of $$4y^9-4y$$

comes out to be $$4y(y^4+1)(y^2+1)(y+1)(y-1)$$

How would you approach to factor?

PS: My math is very rusty.

Answer 1

\begin{align*} 4y^9-4y&=4y(y^8-1)\\ &=4y((y^4)^2-1)\\ &=4y(y^4-1)(y^4+1)\\ &=4y((y^2)^2-1)(y^4+1)\\ &=4y(y^2-1)(y^2+1)(y^4+1)\\ &=4y(y-1)(y+1)(y^2+1)(y^4+1). \end{align*} I am simply using $a^2-b^2=(a-b)(a+b)$. For example, at the second equation we took $a=y^4,\ b=1$.

Answer 2

Factorization of $4y$ is trivial, leaving you with the polynomial

$$y^8-1$$

Then the factorization of any polynomial is the product of the monomials $(y-r_k)$ where $r_k$ are the roots, possibly taken with their multiplicites. So you need to solve

$$y^8=1,$$ or $$"y=\sqrt[8]1".$$

The computation of the eighth roots must be made in the complex and can be carried out using the polar form:

$$y^8=e^{i2k\pi}\iff y=e^{ik\pi/4}.$$

Among the solutions, two are real ($k=0,4$),

$$y=1,y=-1$$

two are pure imaginary (conjugate, $k=2,6$),

$$y=i,y=-i$$

and the four remaining ones are complex (pairwise conjugate, $k=1,3,5,7$)

$$\pm\frac1{\sqrt2}\pm\frac i{\sqrt2}.$$

If you don't want the imaginary/complex numbers to appear, you can keep the quadratic monomials resulting from conjugate pairs:

$$(y-a-ib)(y-a+ib)=y^2-2ay+a^2+b^2.$$

Finally,

$$4y^9-4y=4y(y-1)(y+1)(y^2+1)(y^2-\sqrt2 y+1)(y^2+\sqrt2 y+1).$$