Question

If (2x – 3) is a factor of 6x² + x + a, find the value of a. With this value of a, factorise the given expression.

Answer 1

Let 2x – 3 = 0
then 2x = 3
⇒ x = ${\displaystyle \frac{3}{2}}$
Substituting the value of x in f(x)
f(x) = 6x² + x + a

${\displaystyle f\left(\frac{3}{2}\right)=6\left(\frac{3}{2}\right)+\frac{3}{2}+a}$

= ${\displaystyle 6\times \frac{9}{4}+\frac{3}{2}}$

= ${\displaystyle \frac{27}{2}+\frac{3}{2}+a}$

= ${\displaystyle \frac{30}{2}+a}$
= 15 + a
∴ 2x – 3 is the factor
∴ Remainder = 0
∴ 15 + a = 0
⇒ a = –15
Now f(x) will be 6x² + x – 15
Dividing 6x² + x – 15 by 2x – 3, we get

${\displaystyle 2x-3\text{)}\overline{6x^2+x-15}(3x+5}$
              6x² – 9x
               –    +             
                     10x – 15
                     10x – 15
                     –     +      
                           x       
∴ 6x² + x – 15 = (2x – 3)(3x + 5).