Question

How do you factor completely \[{{x}^{4}}-3{{x}^{2}}-10\]?

Answer 1

Hint: In this problem, we have to find the factors for the given equation. We can see that the given equation is a quartic equation. We can first convert the quartic equation into its quadratic equation by assuming \[{{x}^{2}}=u\]. We can then factorize the resulted quadratic equation to find the factors.

Complete step-by-step answer:
We know that the given quartic equation is,
 \[{{x}^{4}}-3{{x}^{2}}-10\]
 We can replace \[{{x}^{2}}=u\] in the above equation, we get
\[\Rightarrow {{u}^{2}}-3u-10\] …….. (1)
We can now find the factors for the above quadratic equation.
We know that we can find the factors of the quadratic equation by splitting the term with u.
We can split the middle term, as when we add it the coefficient of u and when we multiply it we will get the constant term.
\[\begin{align}
  & \Rightarrow -5+2=-3 \\
 & \Rightarrow -5\times 2=-10 \\
\end{align}\]
Therefore, we can write the equation (1) as,
\[\Rightarrow {{u}^{2}}-5u+2u-10\]
We can now separate the above two terms and take the common terms out, we get
\[\Rightarrow u\left( u-5 \right)+2\left( u-5 \right)\]
we can now write the above step as,
\[\Rightarrow \left( u-5 \right)\left( u+2 \right)\]
We can now replace \[u={{x}^{2}}\], we get
\[\Rightarrow \left( {{x}^{2}}-5 \right)\left( {{x}^{2}}+2 \right)\]
We can now write the first term using the difference of square formula, we get
\[\Rightarrow \left( x-\sqrt{5} \right)\left( x+\sqrt{5} \right)\left( {{x}^{2}}+2 \right)\]
Therefore, the factors of \[{{x}^{4}}-3{{x}^{2}}-10\] are \[\left( x-\sqrt{5} \right)\left( x+\sqrt{5} \right)\left( {{x}^{2}}+2 \right)\].

Note: Students make mistakes while finding the factors, as we have to split the middle term in order to get the sum equals to the middle term itself and product equals to the constant term. We should also remember that, we have to replace the term, which we have assigned initially to get the exact factor of the given equation.