Question

How do you factor the following expression?
${x^5} + {x^3} + {x^2} + 1$

Answer 1

Hint: The given expression is a $5^{th}$ degree polynomial containing only four terms. On close observation, we can see that by taking ${x^3}$ outside from the first two terms, the residual will be the same as the last two terms. Later we can apply the formula $\left( {{a^3} + {b^3}} \right) = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)$ to expand it further.

Complete step-by-step solution:
According to the question, we have been given an algebraic expression and we have to show how we can factorize it.
Let the given expression is denoted by $y$ as shown below:
$ \Rightarrow y = {x^5} + {x^3} + {x^2} + 1$
This is a $5^{th}$ degree polynomial expression. If we closely observe it, we can see that ${x^3}$ is common in the first two terms. So taking it outside, we’ll get:
\[ \Rightarrow y = {x^3}\left( {{x^2} + 1} \right) + \left( {{x^2} + 1} \right)\]
Now again we can see that \[\left( {{x^2} + 1} \right)\] is common throughout the expression. Taking it outside also, we’ll get:
\[ \Rightarrow y = \left( {{x^2} + 1} \right)\left( {{x^3} + 1} \right){\text{ }}.....{\text{(1)}}\]
Further, we know the algebraic formula of $\left( {{a^3} + {b^3}} \right)$ as given below:
$ \Rightarrow \left( {{a^3} + {b^3}} \right) = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)$
Applying this formula for \[\left( {{x^3} + 1} \right)\] in the equation (1), we’ll get:
\[
   \Rightarrow y = \left( {{x^2} + 1} \right)\left( {x + 1} \right)\left( {{x^2} + x + 1} \right) \\
   \Rightarrow y = \left( {x + 1} \right)\left( {{x^2} + 1} \right)\left( {{x^2} + x + 1} \right){\text{ }}.....{\text{(2)}}
 \]

Thus equations (1) and (2) show the two factored forms of expression ${x^5} + {x^3} + {x^2} + 1$.

Additional Information: A $5^{th}$ degree polynomial can have at most five different linear factors thus giving five roots of the polynomial as we know that it can have at most five real roots.
If we generalize this concept, we already know that an $n^{th}$ degree polynomial can have at most n different real roots thus there can be only n different linear factors of any $n^{th}$ degree polynomial.

Note: We have used an algebraic formula of $\left( {{a^3} + {b^3}} \right)$ in the above question. Some other important algebraic formulas are:
$
   \Rightarrow \left( {{a^3} - {b^3}} \right) = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right) \\
   \Rightarrow \left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)
 $