Question

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

Answer 1

Hint:
Here, we will use the concept of the factorization. First, we will convert the given equation into the quadratic equation by substituting the value of the square of the variable as some other variable. This will convert the quartic equation into quadratic equation and then we will split the middle term of the equation. Then we will form the factors by taking the common terms in the equation to get the equation in the factored form.

Complete step by step solution:
Given equation is \[{x^4} - 3{x^2} - 4\].
Factorization is the process in which a number is written in the forms of its small factors which on multiplication give the original number.
First, we will convert the given equation into the quadratic equation by substituting \[{x^2}\] as \[t\] i.e. \[{x^2} = t\]. Therefore, the equation becomes
\[ \Rightarrow {x^4} - 3{x^2} - 4 = {t^2} - 3t - 4\]
Now we will simply form the factors of the quadratic equation by splitting the middle term into two parts such that its multiplication will be equal to the product of the first term and the third term of the equation. Therefore, we get
\[ \Rightarrow {x^4} - 3{x^2} - 4 = {t^2} - 4t + t - 4\]
Now we will be taking \[t\] common from the first two terms and taking 1 common from the last two terms. Therefore the equation becomes
\[ \Rightarrow {x^4} - 3{x^2} - 4 = t\left( {t - 4} \right) + 1\left( {t - 4} \right)\]
Now we will take \[\left( {t - 4} \right)\] common from the equation we get
\[ \Rightarrow {x^4} - 3{x^2} - 4 = \left( {t - 4} \right)\left( {t + 1} \right)\]
Now we will put the value of \[t\] in the above equation. Therefore, we get
\[ \Rightarrow {x^4} - 3{x^2} - 4 = \left( {{x^2} - 4} \right)\left( {{x^2} + 1} \right)\]
We can write the above equation as
\[ \Rightarrow {x^4} - 3{x^2} - 4 = \left( {{x^2} - {2^2}} \right)\left( {{x^2} + 1} \right)\]
We know that \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]. Therefore by using this property in the above equation, we get
\[ \Rightarrow {x^4} - 3{x^2} - 4 = \left( {x + 2} \right)\left( {x - 2} \right)\left( {{x^2} + 1} \right)\]

Hence the factored form of the equation \[{x^4} - 3{x^2} - 4\] is equal to \[\left( {x + 2} \right)\left( {x - 2} \right)\left( {{x^2} + 1} \right)\].

Note:
Here we will note that we should split the middle term very carefully according to the basic condition which is that the middle term i.e. term with the single power of the variable should be divided in such a way that its multiplication must be equal to the product of the first and the last term of the equation. Factors can be the same but in our case it’s different. We should know that the factors we have obtained on solving that we will get the given equation. Factors are the smallest part of the number or equation which on multiplication will give us the actual number of equations. Generally in these types of questions, algebraic identities can be used to solve and make the factors.