Question

Factors of ${a^4} + {a^2} + 1$ are
$\left( a \right){\text{ }}\left( {a - 4} \right)\left( {a + 2} \right)$
$\left( b \right){\text{ }}\left( {{a^2} + 2 - 1} \right)\left( {{a^2} + a + 2} \right)$
$\left( c \right){\text{ }}\left( {{a^2} + a + 1} \right)\left( {{a^2} - a + 1} \right)$
$\left( d \right){\text{ }}\left( {a - 1} \right)\left( {{a^2} - a + 1} \right)$

Answer 1

Hint: For solving this type of question we should first think a bit and try to apply some algebraic formula either by morphing the equation or directly applying. So first of all we will expand the equation so that we are able to use the formula ${a^2} - {b^2} = (a + b)(a - b)$ and then we will get the factors of it.

Formula used:
The algebraic formula used in this question is given by
${a^2} - {b^2} = (a + b)(a - b)$

Complete step-by-step answer:
We have the equation which is given as ${a^4} + {a^2} + 1$ and we have to find the factors of it. So for this, we will use the algebraic formula and will solve it.
So the equation can also be written as
$ \Rightarrow \left( {{a^4} + 2{a^2} + 1} \right) - {a^2}$
And on solving the equation which is in braces, we get
$ \Rightarrow \left[ {{{\left( {{a^2}} \right)}^2} + 2 \times {a^2} + {1^2}} \right] - {a^2}$ , and we will name it equation $1$
Since we know ${(a + b)^2} = {a^2} + {b^2} + 2ab$
So, ${\left( {{a^2}} \right)^2} + 2 \times {a^2} + {1^2}$ can also be written as
$ \Rightarrow {\left( {{a^2} + 1} \right)^2}$
Therefore, the equation $1$ can be written as
$ \Rightarrow {\left( {{a^2} + 1} \right)^2} - {a^2}$
Since we have seen in the formula that ${a^2} - {b^2} = (a + b)(a - b)$
Therefore by using the above formula, the equation can be written as
$ \Rightarrow \left( {{a^2} + a + 1} \right)\left( {{a^2} - a + 1} \right)$
Hence, the option $\left( c \right)$ is correct.

Note: Here the question only asked us to find the factors but if they ask us to find the factorization then we have to solve more and for this, we have to factorize the polynomial into the lower degree. And for this by using the formula ${a^2} - {b^2} = (a + b)(a - b)$ and expanding the equation we will get the other factors. Because in this the equation cannot be in polynomial factorization. So in this way, we can solve it, find it further and check. Also, we have to be cautious while solving this type of question.