Question

vikas has Rs.(x^3+2ax+b), with this money he can buy exactly (x-1) jeans or (x+1) shirts with no money left. How much money vikas has, if x=4?

Answer 1

Vikas has Rs (x³ + 2ax + b) .
Case 1 :- He can buy exactly (x - 1) jeans
means, Rs (x³ + 2ax + b) is divisible by (x - 1)
or you can say that x = 1 is zero of (x³ + 2ax + b)
Now, put x = 1 in x³ + 2ax + b
e.g., 1³ + 2a.1 + b = 0
⇒ 2a + b = -1 --------(1)

Case 2 :- he can also buy exactly (x + 1) shirts
means, Rs (x³ + 2ax + b) is divisible by (x + 1)
or, you can say that x = -1 is zero of (x³ + 2ax + b)
now , put x = -1 in x³ + 2ax + b
e.g., -1 - 2a + b = 0
⇒-2a + b = 1 ------(2)

Solve equations (1) and (2),
2b = 0 ⇒ b = 0 and a = -1/2
So, vikas has Rs. (x³ - x)
Now, at x = 4 , vikas has Rs (4³ - 4) = 60

Hence, answer is Rs. 60

Answer 2

If Vikas can buy exactly x-1 pair of jeans, x³+2ax+b must be exactly divisible by x-1. So, x=1 is a root of x³+2ax+b.
Putting x=1 in the cubic equation, we get 1+2a+b = 0 → 2a+b = -1

If Vikas can buy exactly x+1 shirts, x³+2ax+b must be exactly divisible by x+1. So, x=-1 is a root of x³+2ax+b.
Putting x=-1 in the cubic equation, we get -1-2a+b = 0 → -2a+b = 1

Solving these 2 equations, we get b=0 and a=-1/2
If x=4, the money Vikas has is 4³-4 = 60.
Hence, the answer is 60.