Question

If (2x + 1) is a factor of both the expressions 2x² – 5x + p and 2x² + 5x + q, find the value of p and q. Hence find the other factors of both the polynomials.

Answer 1

Let 2x + 1 = 0, then 2x = –1
x = `-(1)/(2)`
Substituting the value of x in
f(x) = 2x² – 5x + p

`f(-1/2) = 2((-1)/2) –5((-1)/2) + "p"`

= `2 xx (1)/(4) + (5)/(2) + "p"`

= `(1)/(2) + (5)/(2) + "p"`
= 3 + p
∵ 2x + 1 is the factor of p(x)
∴ Remainder = 0
⇒ 3 + p = 0
⇒ p = –3
Again substituting the value of x in q (x)
q(x) = 2x2 + 5x + q

`q(-1/2) = 2(-1/2)^2 + 5(-1/2) + q`

= `2 xx (1)/(4) - (5)/(2) + q`

= `(1)/(2) - (5)/(2) + q`

= `-(4)/(2) + q`
= q – 2
2 x + 1is the factor of q (x)
Remainder = 0
⇒ q – 2 = 0
⇒ q = 2
Hence p = –3, q = 2
Now (i) ∵ 2x + 1 is the factor of p(x)
= 2x² – 5x – 3
∴ Dividing p(x) by 2x + 1,

`2x + 1")"overline(2x^2 - 5x - 3)("x - 3`
             2x² + x
            –      –              
                    –6x – 3
                    –6x  –3
                    +    +      
                         x        
∴ 2x² – 5x – 3 = (2x + 1)(x – 3)
(ii) ∵ 2x + 1 is the factor of q(x) = 2x² + 5x + 2
∴ Dividing q(x) by 2x + 1,

`2x + 1")"overline(2x^2 + 5x + 2)("x + 2`
             2x² + x
            –      –              
                    4x + 2
                    4x + 2
                    –    –     
                         x        
∴ 2x² + 5x + 2 = (2x + 1)(x + 2).