Mathematics 2000 JAMB Past Questions - Question 19: If (x-1), (x+1) and (x-2) are factors of the polynomial ax³ +bx+cx -1, find a,b,c receptively

Correct Answer

A

Explanation

To find the values of a, b, and c, we can use the fact that if (x - 1), (x + 1), and (x - 2) are factors of the polynomial, then when we substitute x = 1, x = -1, and x = 2 into the polynomial, the result should be zero.Let's substitute x = 1 into the polynomial:a(1)³ + b(1) + c(1) - 1 = 0a + b + c - 1 = 0 ----(1)Now, substitute x = -1 into the polynomial:a(-1)³ + b(-1) + c(-1) - 1 = 0-a - b - c - 1 = 0 ----(2)Finally, substitute x = 2 into the polynomial:a(2)³ + b(2) + c(2) - 1 = 08a + 2b + 2c - 1 = 0 ----(3)We now have a system of three equations (equations 1, 2, and 3) with three unknowns (a, b, and c). We can solve this system of equations to find the values of a, b, and c.Adding equations (1) and (2), we get:2b = 0b = 0Substituting b = 0 into equations (1) and (2), we have:a + c - 1 = 0 ----(4)-a - c - 1 = 0 ----(5)Adding equations (4) and (5), we get:0 = 0This means that the equations (4) and (5) are dependent, and we cannot determine the values of a and c independently. This implies that there are infinitely many possible values for a and c that satisfy the given conditions.In conclusion, we can determine that b = 0, but the values of a and c can vary.