Question

Factorise $ 4{x^3} + 20{x^2} + 33x + 18 $ , given that $ \left( {2x + 3} \right) $ is a one factor.

Answer 1

Hint: In this question we have to find the factors of the given polynomial when one factor of the polynomial is already given. In order to solve this question, we divide the given polynomial by the one factor then the quotient obtained would be one of the factors of the polynomial.

Complete step-by-step answer:
Given:
The given polynomial –
 $ 4{x^3} + 20{x^2} + 33x + 18 $
The given factor of the polynomial –
$ \left( {2x + 3} \right) $
Now dividing the given polynomial $ 4{x^3} + 20{x^2} + 33x + 18 $ by the factor $ \left( {2x + 3} \right) $ we have,
$ \dfrac{{4{x^3} + 20{x^2} + 33x + 18}}{{\left( {2x + 3} \right)}} = 2{x^2} + 7x + 6 $
So, the quotient obtained is a factor of the given polynomial.
We can further factorise this value by using the Grouping Method.
First, we split the middle part of this polynomial into two parts in such a way that their sum is equal to the middle part of the polynomial and the product is equal to the product of the first term and last term. So, we have,
 $ 2 \times 6 = 12 $
So, the sum of the two numbers should be 7 and the product should be equal to 12. The numbers are –
$ 4 + 3 = 7 $
And,
 $ 4 \times 3 = 12 $
So, we can write the polynomial $ 2{x^2} + 7x + 6 $ in the following way –
$
\Rightarrow 2{x^2} + 7x + 6 = 2{x^2} + 4x + 3x + 6\\
\Rightarrow 2{x^2} + 7x + 6 = 2x\left( {x + 2} \right) + 3\left( {x + 2} \right)\\
\Rightarrow 2{x^2} + 7x + 6 = \left( {x + 2} \right)\left( {2x + 3} \right)
$
So, the two factors of the polynomial $ 2{x^2} + 7x + 6 $ are $ \left( {x + 2} \right){\rm{ and }}\left( {2x + 3} \right) $ .
We can write all the factors of the polynomial in this form –
$\Rightarrow 4{x^3} + 20{x^2} + 33x + 18 = \left( {2x + 3} \right)\left( {x + 2} \right)\left( {2x + 3} \right) $
Therefore, the factors of the polynomial $ 4{x^3} + 20{x^2} + 33x + 18 $ are –
 $ \left( {2x + 3} \right),\left( {x + 2} \right){\rm{ and }}\left( {2x + 3} \right) $

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 3 therefore the number of the factors of the polynomial is also 3.