Which of the following is a zero for the function f(x) = x^3 – 3x^2 – 7x + 5?

To find a zero for the function f(x) = x³ – 3x² – 7x + 5, we need to solve the equation f(x) = 0.

This means we are looking for values of x that make the equation equal to zero. We can test potential zeros by substituting different values of x into the function.

For example, let’s try some integers:

• For x = 1:
  f(1) = 1³ – 3(1)² – 7(1) + 5 = 1 – 3 – 7 + 5 = -4 (not a zero).
• For x = -1:
  f(-1) = (-1)³ – 3(-1)² – 7(-1) + 5 = -1 – 3 + 7 + 5 = 8 (not a zero).
• For x = 2:
  f(2) = 2³ – 3(2)² – 7(2) + 5 = 8 – 12 – 14 + 5 = -13 (not a zero).
• For x = -2:
  f(-2) = (-2)³ – 3(-2)² – 7(-2) + 5 = -8 – 12 + 14 + 5 = -1 (not a zero).
• For x = 3:
  f(3) = 3³ – 3(3)² – 7(3) + 5 = 27 – 27 – 21 + 5 = -16 (not a zero).
• For x = -3:
  f(-3) = (-3)³ – 3(-3)² – 7(-3) + 5 = -27 – 27 + 21 + 5 = -28 (not a zero).
• For x = 5:
  f(5) = 5³ – 3(5)² – 7(5) + 5 = 125 – 75 – 35 + 5 = 20 (not a zero).

If we checked a range of integers and still did not find a zero, we could use methods like synthetic division or the Rational Root Theorem to find possible rational roots.

In conclusion, testing values may take time, but it’s essential to find zeros of the function. You might also apply numerical methods or graph the function to visually identify where it crosses the x-axis. Remember, zeros are points on the graph where f(x) = 0.