Question

Factor by grouping
 $ 12{x^2} - 16x + 15x - 20 $
(A) $ \left( {4x - 5} \right)\left( {3x + 4} \right) $
(B) $ \left( {12x - 5} \right)\left( {x + 4} \right) $
(C) $ \left( {12x + 5} \right)\left( {x - 4} \right) $
(D) $ \left( {4x + 5} \right)\left( {3x - 4} \right) $

Answer 1

Hint: In this question an algebraic polynomial in the form of a variable x is given and by using the grouping method we have to find the factors of the given polynomial. In the grouping method, we find the common coefficients between the terms in the polynomial and group them into the brackets. Since the highest power of the given polynomial is 2 therefore the number of the factors of this polynomial would also be 2.

Complete step-by-step answer:
Given:
The polynomial given is –
 $ 12{x^2} - 16x + 15x - 20 $
From this polynomial consider first two terms and find the common coefficients we have,
 $\Rightarrow 12{x^2} - 16x = 4x\left( {3x - 4} \right) $
And now finding the common coefficients for the next two terms we have,
 $ 15x - 20 = 5\left( {3x - 4} \right) $
Now grouping the first two and then next two terms of the polynomial by using the common coefficients between them we have -
 $\Rightarrow 12{x^2} - 16x + 15x - 20 = 4x\left( {3x - 4} \right) + 5\left( {3x - 4} \right) $
Now grouping the common terms, we get,
So, we have calculated the two factors for the given polynomial as –
$\Rightarrow 12{x^2} - 16x + 15x - 20 = \left( {3x - 4} \right)\left( {4x + 5} \right) $
Therefore, the factors of the given polynomial are $ \left( {3x - 4} \right) $ and $ \left( {4x + 5} \right) $ .

So, the correct answer is “Option D”.

Note: The number of the factors of a polynomial depends upon the highest power of the polynomial. If the highest power of the polynomial is n then the number of the factors of the polynomial would also be n. For example, if the polynomial is $ a{x^n} + bx + c $ , where a, b and c are arbitrary constants and n is the power of the variable x. So, the number of the factors of this polynomial would be equal to the highest power that is n. Therefore, this polynomial has n number of factors.