Question

The function f is such that f(x) = 4x³ - 8x² + ax + b , where ( a ) and ( b ) are constants. It is given that ( 2x - 1 ) is a factor of f(x) , and when f(x) is divided by ( x + 2 ), the remainder is 20. Find the remainder when f(x) is divided by ( x - 1 ).

A. 18
B. 20
C. 22
D. 24

Answer 1

Using the Remainder Theorem, after finding the values for constants a and b (-16 and 14 respectively), we determine that the remainder when f(x) is divided by (x - 1) is 18.

Step-by-step explanation:

To solve the polynomial division problem, we need to use the Remainder Theorem. According to the Remainder Theorem, if a polynomial f(x) is divided by (x - c), the remainder is f(c).

Since (2x - 1) is a factor of f(x), we can set x = 1/2 to get:

f(1/2) = 0

This gives us:

4(1/2)^3 - 8(1/2)^2 + a(1/2) + b = 0

a/2 + b = 6

If f(x) is divided by (x + 2), and the remainder is 20, we substitute x = -2:

f(-2) = 20

This gives:

4(-2)^3 - 8(-2)^2 - 2a + b = 20

-2a + b = 36

Now we have a system of equations:

• a/2 + b = 6
• -2a + b = 36

Solving the system, we find that a = -16 and b = 14.

Finally, to find the remainder when f(x) is divided by (x - 1), we substitute x = 1 into the original function:

f(1) = 4(1)^3 - 8(1)^2 - 16(1) + 14

The remainder is 18, so the correct answer is Option A.