Question

the function f is defined by f(x)=(x-1)(x+5)(x-2). which of the following is NOT an x-intercept of the graph of y=f(x) in the xy-plane?
(-5, 0)
(1, 0)
(5, 0)
(12, 0)

Answer 1

To find the x-intercepts of the function \( f(x) = (x-1)(x+5)(x-2) \), we need to set \( f(x) = 0 \) and solve for \( x \).

The x-intercepts occur when:

\[
f(x) = 0
\]
which means:
\[
(x-1)(x+5)(x-2) = 0
\]

Setting each factor equal to zero gives us the x-intercepts:

1. \( x - 1 = 0 \) → \( x = 1 \)
2. \( x + 5 = 0 \) → \( x = -5 \)
3. \( x - 2 = 0 \) → \( x = 2 \)

Thus, the x-intercepts are \( x = -5 \), \( x = 1 \), and \( x = 2 \).

Now let's check the given options to see which is NOT an x-intercept:

- \((-5, 0)\) → \( x = -5 \) is an intercept.
- \((1, 0)\) → \( x = 1 \) is an intercept.
- \((5, 0)\) → \( x = 5 \) is NOT an intercept.
- \((12, 0)\) → \( x = 12 \) is NOT an intercept.

Thus, both \((5, 0)\) and \((12, 0)\) are not intercepts, but since the question asks for the answer that is definitely not an x-intercept, the answer is:

\((5, 0)\) is NOT an x-intercept of the graph of \( y = f(x) \).