Question

How do you factor$21{{m}^{2}}-57m+30$?

Answer 1

Hint: To solve these types of questions, first take out the common factor from the given equation, and then by using splitting the middle term or by using cubic and quadratic identities the given expression can be factorized.

Complete step by step answer:
Factorization in general can be defined as writing a mathematical object or expression as a product of several factors that are simpler and smaller. Factoring a large expression into small ones can greatly simplify a mathematical problem and makes it easier to solve mathematical problems.
Given:
 $21{{m}^{2}}-57m+30$
First, let us simplify the expression to easily solve.
We can see that the whole expression is divisible by $3$, therefore dividing the whole equation by $3$, to get:
$\Rightarrow \dfrac{21{{m}^{2}}}{3}-\dfrac{57m}{3}+\dfrac{30}{3}$
Now, simplify the fractions.
$\Rightarrow 7{{m}^{2}}-19m+10$
By applying splitting the middle term method, where we can express the middle term as the sum of the product of the first and the last term, we get:
$\Rightarrow 7{{m}^{2}}-5m-14m+10$
The middle term can be split in the above way as,
$\Rightarrow -5m+\left( -14m \right)=19$
$\Rightarrow -5\times -14=70$
Now, taking common out from each term, we get,
$\Rightarrow m\left( 7m-5 \right)-2\left( 7m-5 \right)$
Now group the factors together.
On further simplifying the above expression and taking the common terms out again we get,
$\Rightarrow \left( 7m-5 \right)\left( m-2 \right)$

Hence, by splitting the middle term we get the factors of $21{{m}^{2}}-57m+30$ as $\left( 7m-5 \right)$$\left( m-2 \right)$

Note: Keep in mind to solve these questions carefully while doing the calculations and opening the parenthesis in the expression. If common factors are not obtained for the pairs, then try again by rearranging the terms and then taking out common terms. Remembering the common and most used algebraic identities and formulas will also help a lot while solving these questions. Apart from using splitting the middle terms to factorize the expression, we can also use algebraic identities as well.