Question

How do you simplify $ - \dfrac{1}{4}m\left( {8{m^2} + m - 7} \right) $ ?

Answer 1

Hint: In the given question, we are required to simplify an algebraic expression given to us in the problem. So, we add and subtract the like terms so as to simplify the algebraic expression. We can use algebraic rules and properties in order to simplify the given algebraic expression. We would factor the cubic expression using the factorisation method.

Complete step by step solution:
We would use the BODMAS rule in order to simplify the algebraic expression $ - \dfrac{1}{4}m\left( {8{m^2} + m - 7} \right) $ . BODMAS is an acronym for the sequence in which the mathematical operations are to be done. In BODMAS, B stands for brackets, O stands for of, D stands for division, M stands for multiplication, A stands addition, S stands subtraction.
So, $ - \dfrac{1}{4}m\left( {8{m^2} + m - 7} \right) $
So, we open up the brackets first and multiply the terms going by sequence set by the acronym BODMAS.
 $ \Rightarrow - \dfrac{1}{4}\left( {8{m^3} + {m^2} - 7m} \right) $
Opening the bracket and simplifying the expression further, we get,
 $ \Rightarrow \left( { - 2{m^3} - \dfrac{1}{4}{m^2} + \dfrac{7}{4}m} \right) $
Hence, the expression $ - \dfrac{1}{4}m\left( {8{m^2} + m - 7} \right) $ can be simplified as $ \left( { - 2{m^3} - \dfrac{1}{4}{m^2} + \dfrac{7}{4}m} \right) $ using the BODMAS rule.
Now, to find the zeros of the cubic polynomial $ \left( { - 2{m^3} - \dfrac{1}{4}{m^2} + \dfrac{7}{4}m} \right) $ , we need to factorise the polynomial. So, first we factor m out out since all the terms consist of m.
 $ \Rightarrow - \dfrac{1}{4}m\left( {8{m^2} + m - 7} \right) $
Now, we need to factor the quadratic expression. So, we would use splitting the middle term method.
 $ \Rightarrow - \dfrac{1}{4}m\left( {8{m^2} + 8m - 7m - 7} \right) $
 $ \Rightarrow - \dfrac{1}{4}m\left( {8m\left( {m + 1} \right) - 7\left( {m + 1} \right)} \right) $
 $ \Rightarrow - \dfrac{1}{4}m\left( {8m - 7} \right)\left( {m + 1} \right) $
So, $ - \dfrac{1}{4}m\left( {8m - 7} \right)\left( {m + 1} \right) $ is the factored and simplified form of the equation $ - \dfrac{1}{4}m\left( {8{m^2} + m - 7} \right) $
So, the correct answer is “ $ - \dfrac{1}{4}m\left( {8m - 7} \right)\left( {m + 1} \right) $”.

Note: The given problem deals with algebraic expression. There is no fixed way of simplifying a given algebraic expression. Care should be taken while doing the calculation steps. Algebraic identities and rules may also be used as and when required as they help simplify complex and tedious tasks.