# If f(x)=24x^3 + px^2 - 5x+q has two factors (2x+1) and (3x-1),the value of p must be?

1
by mindstalker

2015-05-06T09:53:31+05:30
F(x) = 24x^3 + px^2 - 5x + q
factors are 2x + 1 & 3x -1
let us take the factor of 2x + 1 = 0
2x = - 1
x = -1/2
apply the x value in f(x) we get
f(-1/2) = 24(-1/2)^3 + p(-1/2)^2 - 5(-1/2) + q
= 24 (-1/8) + p(1/4) + 5/2 + q
= -3 + p(1/4)  + 5/2+q
= -3 +5/2+ q + p(1/4)
= (-6+5)/2 + q + p (1/4)
=  p(1/4) +q -1/2
p(1/4) + q = 1/2
p + 4q = 4/2
p + 4q = 2      consider this as equ (1)
now take the another factor 3x -1 = 0
3x = 1
x = 1/3
apply the x value in f(x) we get
f(1/3) = 24 ( 1/3) ^3 + p(1/3) ^2 - 5(1/3) + q
= 8 + p(1/3) - 5/3+q
= p(1/3) + q + 8 - 5/3
= p (1/3) + q + (24 -5) /3
= p(1/3) + q + 19 /3
p(1/3) +q = -19/3
p + 3q = ((-19) (3) )/3
p + 3q = -19 consider it as equ (2)
solving (1) & (2) we get ,
p + 4q = 2
p + 3q = - 19 (subtract we get )
....................................
q = 21
apply q value in any one the above equ we get p value
p + 4( 21) = 2
p + 84 = 2
p = 2 -84
p =-82
therefore p is - 84
But the options given for the value of p are 2,1,0,-2