Question

How do you factor completely ${{x}^{4}}-1$?

Answer 1

Hint: Factoring is a process of finding the factors of the given expression with which it is formed. For example, the factors of the arithmetic value 21 are values 3 and 7. These factors when multiplied together will give the value 21. Similarly, we can find the factors of an algebraic expression.

Complete step-by-step answer:
The given algebraic expression is ${{x}^{4}}-1$. This is a polynomial expression with only one variable x. The term polynomial indicates that it has variables, coefficients, and constants.
Here in this expression, we see the variable x and its coefficient 1 and a constant value 1. Also, we can see that the variable is an exponential one and its exponent is equal to 4.
We can always aim to reduce the value of the exponent of the variable in an expression. Let us try to apply this idea in the given expression.
${{x}^{2}}-1={{\left( {{x}^{2}} \right)}^{2}}-1$
If we assume that $a={{x}^{2}}$ and $b=1$, the above equation will be equivalent to ${{a}^{2}}-{{b}^{2}}$. We know that the algebraic identity ${{a}^{2}}-{{b}^{2}}$ is equal to $\left( a+b \right)\left( a-b \right)$. Therefore
${{x}^{2}}-1=\left( {{x}^{2}}+1 \right)\left( {{x}^{2}}-1 \right)$
Similarly, $\left( {{x}^{2}}-1 \right)$ can be written as $\left( x+1 \right)\left( x-1 \right)$.
${{x}^{2}}-1=\left( {{x}^{2}}+1 \right)\left( x+1 \right)\left( x-1 \right)$ ……(1)
Now the only remaining non-factored component is $\left( {{x}^{2}}+1 \right)$. Let us check the root of this particular component.
${{x}^{2}}+1=0$
.$\Rightarrow {{x}^{2}}=-1$
$\Rightarrow x=\sqrt{-1}=i$
The value $i=\sqrt{-1}$ is an imaginary number. Because this situation implies that the variable x multiplied by itself will give a value equal to -1, which is a negative number. This is not possible while working with real numbers. Hence, we can conclude that the component $\left( {{x}^{2}}+1 \right)$ cannot be factored by using real numbers.
Therefore the completely factored form of the expression ${{x}^{4}}-1$ is equal to $\left( {{x}^{2}}+1 \right)\left( x+1 \right)\left( x-1 \right)$.

Note: If we choose to factorize the expression further, we can do it by using imaginary numbers. As a result, the factored expression will contain complex numbers. As we already saw, $i=\sqrt{-1}$ or ${{i}^{2}}=-1$. Hence $\left( {{x}^{2}}+1 \right)$ can be written as $\left( {{x}^{2}}-{{i}^{2}} \right)$.
${x^2} - 1 = \left( {{x^2} - {i^2}} \right)\left( {x + 1} \right)\left( {x - 1} \right)$
${x^2} - 1 = \left( {x + i} \right)\left( {x - i} \right)\left( {x + 1} \right)\left( {x - 1} \right)$
The above equation is the completely factored form of the given expression.