Question

Find the Factors of ${x^4} - {x^2} - 12$.

Answer 1

Hint:
Assume ${x^2}$ to be t then the equation will be ${t^2} - t - 12$ for factoring we will split -1 in such that multiplication of 2 terms will be equal to -12. ${t^2} - t - 12$ will be equal to $\left( {t - 4} \right)\left( {t + 3} \right)$. Substitute t= ${x^2}$ then we can see the $\left( {{x^2} - 4} \right)$ can be further factories on $\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$. Then we will get$\left( {{x^2} - 4} \right) = \left( {x - 2} \right)\left( {x + 2} \right)$.

Complete step by step solution:
Given, ${x^4} - {x^2} - 12$
Assume ${x^2}$= t equation will be
 $ \Rightarrow {t^2} - t - 12$
 split -1 in such that multiplication of 2 terms will be equal to -12
$ \Rightarrow {t^2} - 4t + 3t - 12$
Taking common t and 3
$ \Rightarrow t\left( {t - 4} \right) + 3\left( {t - 4} \right)$
Now taking common $\left( {t - 4} \right)$
$ \Rightarrow \left( {t - 4} \right)\left( {t + 3} \right)$
Then substitute ${x^2}$= t
$ \Rightarrow \left( {{x^2} - 4} \right)\left( {{x^2} + 3} \right)$
It is known that $\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$ then $\left( {{x^2} - 4} \right) = \left( {x - 2} \right)\left( {x + 2} \right)$
$ \Rightarrow \left( {x - 2} \right)\left( {x + 2} \right)\left( {{x^2} + 3} \right)$

Note:
If $a{x^2} + bx + c$ is a quadratic equation then $\left( {x - h} \right)\left( {x - r} \right)$ then $h + r = \dfrac{{ - b}}{a}$ and $h \times r = \dfrac{{ - c}}{a}$. formula used $\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$.