Question

Factorise $ 6{x^2} + 5x - 6 $
(A) $ \left( {3x + 2} \right)\left( {2x - 3} \right) $
(B) $ \left( {3x - 2} \right)\left( {2x + 3} \right) $
(C) $ \left( {4x + 3} \right)\left( {3x - 4} \right) $
(D) $ \left( {4x - 3} \right)\left( {3x + 4} \right) $

Answer 1

Hint: In this question we have to find the factors of the given polynomial which has three terms and the highest value of power in these terms is 2. We apply the Grouping Method step by step in this polynomial and find the factors required. We start by finding the common coefficients between the terms in the polynomial and group them together in the brackets.

Complete step-by-step answer:
Given:
The polynomial given is –
 $ 6{x^2} + 5x - 6 $
The first term of the polynomial is $ 6{x^2} $ and its coefficient is $ 6 $ .
The middle term of the polynomial is $ 5x $ and its coefficient is $ 5 $ .
And, the last term of the polynomial is a constant and its value is $ - 6 $ .
First, we find the product of the coefficient of the first term and the constant value. So, we get,
 $ 6 \times - 6 = - 36 $
Then we have to find the two factors of $ - 36 $ in such a way that their sum equals the value of the coefficient of the middle term which is $ 5 $ . So, the two factors are –
 $ 9 + \left( { - 4} \right) = 5 $
 $ 9 $ and $ - 4 $ are the two factors of $ - 36 $ .
Now we can rewrite the polynomial using these values in this form –
 $ \Rightarrow 6{x^2} + 5x - 6 = 6{x^2} + 9x - 4x - 6 $
So, taking common value from the first two terms of the polynomial we get,
 $ \Rightarrow 6{x^2} + 9x = 3x\left( {2x + 3} \right) $
Similarly, taking common value from the last two terms of the polynomial we get,
 $ \Rightarrow - 4x - 6 = - 2\left( {2x + 3} \right) $
Now grouping the common terms from all four terms we get,
 $ \Rightarrow \left( {3x - 2} \right)\left( {2x + 3} \right) $
So, the two factors of the given polynomial are –
 $ \Rightarrow 6{x^2} + 5x - 6 = \left( {3x - 2} \right)\left( {2x + 3} \right) $

Note: It should be noted that if the value of the highest power of the polynomial is n then the number of the factors of the polynomial would also be n. For example, in this question the value of the highest power of the polynomial is 2 and the number of the factors of the polynomial is also 2.