Question

How many x-intercepts does the function f(x)=3(2x−1)(x+2)(8x−5)2
have when graphed? (1 point)
The function f(x)
has
x-intercepts when graphed.

Answer 1

To find the number of x-intercepts of the function \( f(x) = 3(2x - 1)(x + 2)(8x - 5)^2 \), we need to determine the values of \( x \) that make \( f(x) = 0 \).

The function \( f(x) \) is equal to zero when any of its factors are zero. We will set each factor equal to zero:

1. \( 2x - 1 = 0 \)
Solving for \( x \):
\[
2x = 1 \\
x = \frac{1}{2}
\]

2. \( x + 2 = 0 \)
Solving for \( x \):
\[
x = -2
\]

3. \( (8x - 5)^2 = 0 \)
Solving for \( x \):
\[
8x - 5 = 0 \\
8x = 5 \\
x = \frac{5}{8}
\]
Note that since this term is squared, the root \( x = \frac{5}{8} \) only counts as one x-intercept, even though it is a double root.

Now we summarize the x-intercepts we found:

- \( x = \frac{1}{2} \)
- \( x = -2 \)
- \( x = \frac{5}{8} \)

Each of these values represents an x-intercept. Therefore, the total number of distinct x-intercepts is:

1. \( \frac{1}{2} \)
2. \( -2 \)
3. \( \frac{5}{8} \)

Thus, the function \( f(x) \) has **3 x-intercepts** when graphed.