Question

Factorise: $ 6{x^2} - 11x + 3 $
(A) $ x = \dfrac{1}{3}{\rm{ and }}\dfrac{2}{3} $
(B) $ x = \dfrac{1}{3}{\rm{ and }}\dfrac{{ - 3}}{2} $
(C) $ x = \dfrac{{ - 1}}{3}{\rm{ and }}\dfrac{3}{2} $
(D) $ x = \dfrac{1}{3}{\rm{ and }}\dfrac{3}{2} $

Answer 1

Hint: In this question a quadratic polynomial in form having one single variable x is given and we have to find the factors of the given polynomial. In order to find the factors of this polynomial we use the Grouping Method. In the grouping method, we find the common coefficients between the terms in the polynomial and group them into the brackets.

Complete step-by-step answer:
Given:
The polynomial given is –
 $ 6{x^2} - 11x + 3 $
The first term of the polynomial is $ 6{x^2} $ and its coefficient is $ 6 $ .
The middle term of the polynomial is $ - 11x $ and its coefficient is $ - 11 $ .
And, the last term of the polynomial is a constant and its value is $ 3 $ .
First, we find the product of the coefficient of the first term and the constant value. So, we get,
 $ 6 \times 3 = 18 $
Then we have to find the two factors of $ 18 $ in such a way that their sum equals the value of the coefficient of the middle term which is $ - 11 $ . So, the two factors are –
 $ - 9 + \left( { - 2} \right) = - 11 $
 $ - 9 $ and $ - 2 $ are the two factors of $ 18 $ .
Now we can rewrite the polynomial using these values in this form –
 $ \Rightarrow 6{x^2} - 11x + 3 = 6{x^2} - 9x - 2x - 3 $
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 - 2x + 3 = - 1\left( {2x - 3} \right) $
Now grouping the common terms from all four terms we get,
 $ \Rightarrow \left( {3x - 1} \right)\left( {2x - 3} \right) $
So, we can write the two factors of the polynomial in the following form –
 $
\Rightarrow \left( {3x - 1} \right) = 0\\
\Rightarrow x = \dfrac{1}{3}
 $
And
 $
\left( {2x - 3} \right) = 0\\
x = \dfrac{3}{2}
 $
Therefore, the two factors of the polynomial are $ x = \dfrac{1}{3} $ and $ x = \dfrac{3}{2} $ .

So, the correct answer is “Option D”.

Note: It should be noted that the number of the factors of a polynomial depends upon the highest power of the polynomial. So, if 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 highest power of the polynomial is 2 and the number of the factors of the polynomial is also 2.