Question

Find the factors of $ {a^3} + a - 3{a^2} - 3 $ .

Answer 1

Hint: In the given problem, we will find factors of $ {a^3} + a - 3{a^2} - 3 $ by grouping method. Observe the first two terms and make the one group of those terms if they have a common factor. Similarly, make the one group of last two terms if they have a common factor.

Complete step-by-step answer:
In this problem, we have to find factors of $ {a^3} + a - 3{a^2} - 3 $ . Observe all terms and if there is any common factor among all terms then take it common out. But here we can see that there is no common factor among all terms. Now observe the first two terms. They have a common factor $ a $ . So, we can write $ {a^3} + a = a\left( {{a^2} + 1} \right) $ . Now observe the last two terms. They have a common factor $ - 3 $ . So, we can write $ - 3{a^2} - 3 = - 3\left( {{a^2} + 1} \right) $ . Hence, to find the required factors, we will make one group of the first two terms and one group of the last two terms (take minus sign common) from the given expression. So, we can write
 $ {a^3} + a - 3{a^2} - 3 $
 $ = \left( {{a^3} + a} \right) - \left( {3{a^2} + 3} \right) $
 $ = a\left( {{a^2} + 1} \right) - 3\left( {{a^2} + 1} \right) $
Now in the above expression we can see that $ \left( {{a^2} + 1} \right) $ is a common factor. So, we can write
 $\Rightarrow {a^3} + a - 3{a^2} - 3 = \left( {{a^2} + 1} \right)\left( {a - 3} \right) $
Hence, we can say that there are two factors
$\Rightarrow \left( {{a^2} + 1} \right) $ and $ \left( {a - 3} \right) $ of $ {a^3} + a - 3{a^2} - 3 $ .

Note: In the given problem, also we can find the factors of $ {a^3} + a - 3{a^2} - 3 $ by regrouping method. Let us rearrange the given expression. So, we can write $ {a^3} - 3{a^2} + a - 3 $ . Observe the first two terms so we can say that they have a common factor $ {a^2} $ . Observe the last two terms so we can say that they have a common factor $ 1 $ . Hence, we will make one group of the first two terms and one group of last two terms from the given expression. This is called a regrouping method. We can use the grouping or regrouping method to find the factors of polynomials if the common factor exists between the groups.