Question

Factorize: $ {x^4} + {x^2} + 1 $

Answer 1

Hint: Use the formulae \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\]which states that the square of the sum of two numbers can be given as the sum of the squares of the individual numbers added to twice the product of those two numbers. Also, the formula $ {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) $ states that the difference between the squares of the two numbers is equal to the product of the sum and difference of those two numbers.

Complete step-by-step answer:
The given expression is $ {x^4} + {x^2} + 1 $ .
We know that, \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\].
To factorise the given equation, we need to calculate the value of \[{\left( {{x^2} + 1} \right)^2}\].
We should use the formula \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\] to find the value of \[{\left( {{x^2} + 1} \right)^2}\] easily.
After comparing \[{\left( {{x^2} + 1} \right)^2}\] with \[{\left( {a + b} \right)^2}\] to find the value of $ a,b $ , we get $ a = {x^2} $ and $ b = 1 $ .
Now we have to substitute the value of $ a = {x^2} $ and $ b = 1 $ in the equation \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\].
\[
\Rightarrow {\left( {{x^2} + 1} \right)^2} = {\left( {{x^2}} \right)^2} + {1^2} + 2 \cdot {x^2} \cdot 1\\
\Rightarrow {\left( {{x^2} + 1} \right)^2} = {x^4} + 2{x^2} + 1......\left( 1 \right)

\]
  $
\Rightarrow {x^4} + {x^2} + 1 = {x^4} + 2{x^2} - {x^2} + 1\\
 = {x^4} + 2{x^2} + 1 - {x^2}\\
 = \left( {{x^4} + 2{x^2} + 1} \right) - {x^2}
 $
Now, to further factorise the given expression use the equation (1). Substitute equation (1) in the equation
 $ \Rightarrow {x^4} + {x^2} + 1 = \left( {{x^4} + 2{x^2} + 1} \right) - {x^2} $
  $
\Rightarrow {x^4} + {x^2} + 1 = \left( {{x^4} + 2{x^2} + 1} \right) - {x^2}\\
 = {\left( {{x^2} + 1} \right)^2} - {x^2}
 $
The form of the expression $ {\left( {{x^2} + 1} \right)^2} - {x^2} $ is related to $ {a^2} - {b^2} $ .
Now, $ {\left( {{x^2} + 1} \right)^2} - {x^2} $ and $ {a^2} - {b^2} $ are to be compared.
After comparing, we get, $ a = {x^2} + 1 $ and $ b = x $ .
Now, to further factorise the given expression use the formula
 $ \Rightarrow {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) $ . Substitute the values of $ a = {x^2} + 1 $ and $ b = x $ in the formula $ {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) $ .
 $ \Rightarrow {\left( {{x^2} + 1} \right)^2} - {x^2} = \left( {{x^2} + 1 + x} \right)\left( {{x^2} + 1 - x} \right)......\left( 2 \right) $
After substituting the equation (2) in the equation $ {x^4} + {x^2} + 1 = {\left( {{x^2} + 1} \right)^2} - {x^2} $ , we get,
  $
\Rightarrow {x^4} + {x^2} + 1 = \left( {{x^2} + 1 + x} \right)\left( {{x^2} + 1 - x} \right)\\
 = \left( {{x^2} + x + 1} \right)\left( {{x^2} - x + 1} \right)
 $
Therefore, the final answer is $ {x^4} + {x^2} + 1 = \left( {{x^2} + x + 1} \right)\left( {{x^2} - x + 1} \right) $

Additional Information:
Factorisation of polynomials is done by separating it in the product of factors which cannot be reduced further. In 1793, Theodor von Schubert produced the first algorithm for the factorization of the polynomial. The linear factors are being deduced from the polynomials in order to factorise the polynomials.
There are some of the important identities which are mostly used to either factorise of an expression.
 $ \Rightarrow {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab $
The square of the sum of two numbers can be given as the sum of the squares of the individual numbers added to twice the product of those two numbers.
 $ \Rightarrow {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab $
The square of the difference of two numbers can be given by subtracting twice the product of the numbers from the sum of the squares of the individual numbers.
 $ \Rightarrow {a^2} - {b^2} = \left( {a + b} \right) \cdot \left( {a - b} \right) $
The difference between the squares of the two numbers is equal to the product of the sum and difference of those two numbers.

Note: In this type of question, special care should be taken with respect to the operations like addition, subtraction, multiplication. There shouldn’t be any mistake while arranging the symbols. BODMAS i.e. the rule of operations should be followed, where first bracket, then division, multiplication, addition and then subtraction should be performed at the last.