Question

For what value of a the polynomial 2x^3+ax^2+11 x+a+3 is exactly divisible by 2x-1

Answer 1

2x - 1 must be a factor of the given polynomial.
One of the roots must be x = 1/2 

Plug in x = 1/2, and equate the polynomial to zero: 
2(1/2)^3 + a(1/2)^2 + 11(1/2) + a + 3 = 0 
1/4 + a/4 + 11/2 + a + 3 = 0 
5a/4 + 35/4 = 0 
5a + 35 = 0 
a = -7 

SOLUTION: 2x^3 - 7x^2 + 11x - 4 is exactly divisible by 2x - 1 

Check: (x^2 - 3x + 4)(2x - 1) = 2x^3 - 7x^2 + 11x - 4 

Answer 2

Given,  2x-1 = 0 i.e x = 1/2   ------------- (i)

Substitute x value in polynomial equation, we get

= 2(1/8) + a(1/4) + 11(1/2) + a + 3 = 0 

= 1/4 + a/4 + 11/2 + a + 3 = 0 

= 1/4 + a/4 + 22/4 + 4a/4 + 12/4 = 0 

= 5a/4 + 35/4 = 0

= 5a + 35 = 0 

a = -7.


Hope this helps!