Question

Factorize: ${{x}^{3}}-3{{x}^{2}}-9x-5$.

Answer 1

Hint: Factorization 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 solution:
The given algebraic expression is ${{x}^{3}}-3{{x}^{2}}-9x-5$. This is a third-degree polynomial expression with a single variable.
In algebra, the grouping is a process of separating the components into equal groups. We can write the given expression in a way that we can group the common components.
$\Rightarrow {{x}^{3}}-3{{x}^{2}}-9x-5={{x}^{3}}+{{x}^{2}}-4{{x}^{2}}-4x-5x-5$
In the above equation, we have written the term $3{{x}^{2}}$ as ${{x}^{2}}-4{{x}^{2}}$ and the term $\;9x$ as $4x-5x$. By doing this, we can make the coefficients of the terms similar. Now, let us group the components according to the common factors.
$\Rightarrow {{x}^{3}}+{{x}^{2}}-4{{x}^{2}}-4x-5x-5={{x}^{2}}\left( x+1 \right)-4\left( x+1 \right)-5\left( x+1 \right)$
The term $\left( x+1 \right)$ is the common factor in the above equation.
$\Rightarrow {{x}^{2}}\left( x+1 \right)-4\left( x+1 \right)-5\left( x+1 \right)=\left( x+1 \right)\left( {{x}^{2}}-4x-5 \right)$
The expression $\left( {{x}^{2}}-4x-5 \right)$ is a second-degree quadratic expression. Let us rearrange the equation.
$\Rightarrow \left( x+1 \right)\left( {{x}^{2}}-4x-5 \right)=\left( x+1 \right)\left( {{x}^{2}}-5x+x-5 \right)$
$\Rightarrow \left( x+1 \right)\left( {{x}^{2}}-5x+x-5 \right)=\left( x+1 \right)\left( x\left( x-5 \right)+1\left( x-5 \right) \right)$
In the above equation, $\left( x-5 \right)$ is the common factor.
$\Rightarrow \left( x+1 \right)\left( x\left( x-5 \right)+1\left( x-5 \right) \right)=\left( x+1 \right)\left( x+1 \right)\left( x-5 \right)$

Therefore $\left( x+1 \right)\left( x+1 \right)\left( x-5 \right)$ is the factored form of the expression ${{x}^{3}}-3{{x}^{2}}-9x-5$.

Note:
While factoring a polynomial expression, a general approach is to decrease the degree of the polynomial. To achieve this, we try to regroup the terms of the expression separately to take out the common factors. In this way, we can reduce a higher degree polynomial into a lower one.