Question

How do you factor ${x^4} + {x^2} + 1$ ?

Answer 1

Hint: In the given question they have given an equation that is ${x^4} + {x^2} + 1$ to find the factors. Whenever we need to find the factors first we need to split or break the mid-term that is ${x^2}$ of the given equation so that by addition or subtraction of the values we get the original value. After splitting the term by taking common terms and simplifying the equation we arrive at the required answer.

Complete step by step answer:
Here in the given question, they have given an equation that is ${x^4} + {x^2} + 1$ to find the factors. Whenever we need to find the factors of any given equation, first we need to split or break the middle term of the equation that is ${x^2}$ of the given equation so that by addition or subtraction of the values, we get the original value.
Now we split the middle term so that we can simplify the equation.
Given, ${x^4} + {x^2} + 1$
Now, if we split the above equation, we get
$ \Rightarrow {x^4} + {x^2} + 1 = {x^4} + 2{x^2} - {x^2} + 1$ (Here if we do $2{x^2} - {x^2}$ we get the same as that in original equation)
Now, by looking at the above expression we try to write the above expression even simpler so that we can get the factors easily.
By rewriting the above expression, we get
$ \Rightarrow {x^4} + {x^2} + 1 = {\left( {{x^2}} \right)^2} + 2\left( {{x^2}} \right)\left( 1 \right) + {1^2} - {x^2}$
In the above expression ${\left( {{x^2}} \right)^2} + 2\left( {{x^2}} \right)\left( 1 \right) + {1^2}$ is the expanded form of ${\left( {{x^2} + 1} \right)^2}$ that is from the general form given by: ${(a + b)^2} = {a^2} + 2ab + {b^2}$ .
Therefore, the above expression is written as
$ \Rightarrow {x^4} + {x^2} + 1 = {\left( {{x^2} + 1} \right)^2} - {x^2}$
Again, the above expression represents the form difference of square that is ${a^2} - {b^2} = (a + b)(a - b)$ where $a = \left( {{x^2} + 1} \right)$ and $b = x$ in this problem. Now, by expanding the above expression we get the factors of the given equation.
Therefore, we get
$ \Rightarrow {x^4} + {x^2} + 1 = \left( {{x^2} + 1 - x} \right)\left( {{x^2} + 1 + x} \right)$

Therefore the factors of ${x^4} + {x^2} + 1$ are $\left( {{x^2} + 1 - x} \right)$ and $\left( {{x^2} + 1 + x} \right)$.

Note:
In this type of question, whenever we split the middle term which is the very important step in this type of problem, if you fail to split the term then you end up with the wrong answer. After splitting try to simplify the expression as much as possible to get the correct factor.