Question

How do you factor $12{{x}^{3}}+2{{x}^{2}}-30x-5$?

Answer 1

Hint: To find the factors of the given equation, we have to use grouping. So, firstly we need to group the first two terms and the next two terms separately from the given equation then taking the greatest common factor from the two groups we get the required factors.

Complete step by step answer:
According to the question, we have been asked to find the factors of $12{{x}^{3}}+2{{x}^{2}}-30x-5$.
To solve the question, we will start with grouping the first two terms and then the next two.
We can say that on grouping the first two terms, we will get
$\Rightarrow \left( 12{{x}^{3}}+2{{x}^{2}} \right)-30x-5$
Similarly on grouping the next two terms by taking negative sign common, we will get
$\Rightarrow \left( 12{{x}^{3}}+2{{x}^{2}} \right)-\left( 30x+5 \right)$
Now, we can see that from $12{{x}^{3}}+2{{x}^{2}}$, we can take $2{{x}^{2}}$ common and from $30x+5$, we can take 5 common. Therefore we can further write the equation as
$\Rightarrow 2{{x}^{2}}\left( 6x+1 \right)-5\left( 6x+1 \right)$
From the above equation, we can see that (6x+1) can be taken out as common. Therefore, we can write
$\Rightarrow \left( 2{{x}^{2}}-5 \right)\left( 6x+1 \right)$
Therefore, we get the value of $12{{x}^{3}}+2{{x}^{2}}-30x-5$ as $\left( 2{{x}^{2}}-5 \right)\left( 6x+1 \right)$.

Hence, we can say that the factors of given expression $12{{x}^{3}}+2{{x}^{2}}-30x-5$ are $\left( 2{{x}^{2}}-5 \right)$ and $\left( 6x+1 \right)$.

Note: The possible mistake one can make while solving the question is not taking negative signs common at the time of grouping last two terms and then ending up with the wrong answer. Also, we can verify our answer by multiplying the obtained factors with each other to see whether we have found the correct answer or not.