Question

Let (x) = 2x^3 − 9x^2 / (13x − 6), and (x) = x − 2. Use the remainder theorem to find:

a) f(2)

b) f(-2)

c) g(4)

d) g(-4)

Answer 1

By substituting the given x-values into the functions f(x) and g(x), we find that f(2) = −1, f(-2) = 13/8, g(4) = 2, and g(-4) = −6.

Step-by-step explanation:

To answer the student's question using the remainder theorem, we need to substitute the given x-values into the given functions and calculate the resulting values.

• For f(2), substitute x = 2 into the function f(x) = 2x³ − 9x² / (13x − 6):
  
  2(2)³ − 9(2)² / (13(2) − 6) = 2(8) − 9(4) / (26 − 6) = 16 − 36 / 20 = −20 / 20 = −1.
• For f(-2), substitute x = -2 into the function f(x):
  
  2(-2)³ − 9(-2)² / (13(-2) − 6) = 2(-8) − 9(4) / (-26 − 6) = −16 − 36 / −32 = −52 / −32 = 13/8.
• For g(4), substitute x = 4 into the function g(x) = x − 2:
  
  4 − 2 = 2.
• For g(-4), substitute x = -4 into the function g(x):
  
  −4 − 2 = −6.

Therefore, the answers are f(2) = −1, f(-2) = 13/8, g(4) = 2, and g(-4) = −6.