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How do you factor ${{x}^{3}}+3{{x}^{2}}-2x-6$?

seo-qna
Last updated date: 13th Jun 2024
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Answer
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Hint: In this problem we have to calculate the factors of the given equation. We can observe that the given equation is the cubic equation. Now we will consider the first two terms individually and take ${{x}^{2}}$ as common. After that we will consider the last two terms and take $-2$ as common. Now we will observe the obtained equation and take appropriate terms as common to get the factors of the given equation.

Complete step by step answer:
Given equation, ${{x}^{3}}+3{{x}^{2}}-2x-6$.
Considering the first two terms. We have the first term ${{x}^{3}}$ and the second term $3{{x}^{2}}$. By observing the above two terms we can take ${{x}^{2}}$ as common. So, taking ${{x}^{2}}$ as common from the first two terms of the given equation, then we will get
$\Rightarrow {{x}^{3}}+3{{x}^{2}}-2x-6={{x}^{2}}\left( x+3 \right)-2x-6$
Considering the last two terms of the given equation. We have last term $-6$ and the third term $-2x$. By observing the above two terms we can take $-2$ as common. So, taking $-2$ as common from the last two terms of the given equation, then we will get
$\Rightarrow {{x}^{3}}+3{{x}^{2}}-2x-6={{x}^{2}}\left( x+3 \right)-2\left( x+3 \right)$
In the above equation we can observe that we can take $x+3$ as common. So, taking $x+3$ as common from the above equation, then we will get
$\Rightarrow {{x}^{3}}+3{{x}^{2}}-2x-6=\left( x+3 \right)\left( {{x}^{2}}-2 \right)$

Hence the factors of the given equation ${{x}^{3}}+3{{x}^{2}}-2x-6$ are ${{x}^{2}}-2$, $x+3$.

Note: In the given equation we have the common factors of the terms in the equation, so we have taken common from the terms and simplify them to get the factors. But in some cases, we don’t have any common factors for the terms in the equation, then the above used method is not applicable. Then we will consider the given polynomial as a function and we will calculate the value of $x$ where $f\left( x \right)=0$ by substituting the random values of $x$. After getting the value of $x$ as $x=a$ we will divide the given equation with $x-a$. Then we will get a quadratic equation as a quotient. Now we will factorise the quadratic equation to get another two factors. After finding the factors of the quadratic equation, the factors of the given cubic polynomial are $x-a$, the factors of the quadratic equation.