I am refreshing my mind on how to factor quadratic equations (or polynomials) as I have not done so in $2$ years and I am required to do so with the problem I was assigned. With the equation below, the first variable (called the coefficient if I remember correctly) has an exponent.
$$k^3 + 4k + 2$$
How would I be able to factor this even if it's not rational per say?
There are a few ways factor Cubic equations, such as the cubic identities, where:
$1.$ $a^3+b^3=(a+b)(a^2-ab+b^2)$
$2.$ $a^3-b^3=(a-b)(a^2+ab+b^2)$
or there is another way, where you use the Factor theorem, that states that if $f(c)=0$ for some c, the $x-c$ is a factor of $f(x)$
The factors of $2: 1,2,-1,-2$
If we substitute $1$ into the equation: $1+4+2 \neq 0$,
If we substitute $-1$ into the equation: $-1-4+2 \neq 0$
If we substitute $-2$ into the equation: $-8-8+2 \neq 0$
And if we substitute $2$ into the equation: $8+8+2 \neq 0$,
Therefore, you simply cannot factorise the equation $k^3+4k+2$.
Hope this helps.