Question

How do you factor ${{x}^{4}}+3{{x}^{2}}+4$ ?

Answer 1

Hint: Here for answering this question we need to find the factors of the given expression ${{x}^{4}}+3{{x}^{2}}+4$ . For doing that let us assume ${{x}^{2}}=a$ and by substituting this value we will get a quadratic expression.

Complete step by step solution:
Now considering from the question we have been asked to find the factors of the given expression ${{x}^{4}}+3{{x}^{2}}+4$ .
For doing that we will assume ${{x}^{2}}=a$ .
Now by substituting this value in the given expression we will come across the following expression given as $\Rightarrow {{a}^{2}}+3a+4$ .
Now we can write this expression simply as $\Rightarrow {{a}^{2}}+4a-a+4={{\left( a+2 \right)}^{2}}-a$ .
Now by further simplifying we will have $\Rightarrow {{\left( {{x}^{2}}+2 \right)}^{2}}-{{x}^{2}}=\left( {{x}^{2}}+2-x \right)\left( {{x}^{2}}+2+x \right)$ .
Hence we can conclude that the factors of the given expression are $\left( {{x}^{2}}+2\pm x \right)$ .

Note: During answering questions of this type we should be sure with our concepts that we apply and the calculations we perform. Very few mistakes are possible in questions of this type and this type of questions can be answered in a short span of time if we have a strong concept. In order to answer this type of question with ease we should practice more questions. This question can also be answered in another way. We can use the formula for finding the roots of the quadratic expression in the form of $a{{x}^{2}}+bx+c$ which is mathematically given as $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . If the roots are ${{x}_{1}},{{x}_{2}}$ then $\left( x-{{x}_{1}} \right),\left( x-{{x}_{2}} \right)$ will be the factors of the expression. So by applying this concept here we will have
$\begin{align}
  & \Rightarrow \dfrac{-3\pm \sqrt{{{3}^{2}}-4\left( 4 \right)}}{2}=\dfrac{-3\pm \sqrt{9-16}}{2} \\
 & \Rightarrow \dfrac{-3\pm i\sqrt{7}}{2} \\
\end{align}$ . Now the factors will be $\left( {{x}^{2}}-\dfrac{-3\pm i\sqrt{7}}{2} \right)$ .