Given that the remainder when f(x) = x3 − 2x2 − ax + b is divided by x − 1is 96 and that x − 4 is a factor, determine the values of a and b.

To solve this problem, we will use the Remainder Theorem and the Factor Theorem.

Step 1: Use the Remainder Theorem The Remainder Theorem states that if a polynomial f(x) is divided by x - k, the remainder is f(k). In this case, we know that the remainder when f(x) is divided by x - 1 is 96. Therefore, we can set up the equation f(1) = 96.

Substitute x = 1 into the equation f(x) = x^3 - 2x^2 - ax + b to get:

f(1) = 1^3 - 21^2 - a1 + b = 96 Simplify to get: -a + b = 98

Step 2: Use the Factor Theorem The Factor Theorem states that if x - k is a factor of f(x), then f(k) = 0. In this case, we know that x - 4 is a factor of f(x). Therefore, we can set up the equation f(4) = 0.

Substitute x = 4 into the equation f(x) = x^3 - 2x^2 - ax + b to get:

f(4) = 4^3 - 24^2 - a4 + b = 0 Simplify to get: -4a + b = -48

Step 3: Solve the system of equations Now we have a system of two equations: -a + b = 98 and -4a + b = -48. We can solve this system to find the values of a and b.

Subtract the second equation from the first to get: 3a = 146, so a = 146/3 = 48.67 (approx)

Substitute a = 48.67 into the first equation to get: -48.67 + b = 98, so b = 98 + 48.67 = 146.67 (approx)

Therefore, the values of a and b are approximately 48.67 and 146.67, respectively.