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 graphe

Answer 1

To determine the number of x-intercepts of the function \( f(x) = 3(2x - 1)(x + 2)(8x - 5)^2 \), we need to identify the roots of the function, which occur when \( f(x) = 0 \).

The function will be zero when any of the factors is zero:

1. **From the factor \( 2x - 1 \)**:
\[
2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}
\]

2. **From the factor \( x + 2 \)**:
\[
x + 2 = 0 \implies x = -2
\]

3. **From the factor \( (8x - 5)^2 \)**:
Since this factor is squared, it will also give us roots when:
\[
8x - 5 = 0 \implies 8x = 5 \implies x = \frac{5}{8}
\]
However, because this factor is squared, it will only count as one unique x-intercept.

Now, let's summarize the x-intercepts:

- The intercept from \( 2x - 1 \) gives \( x = \frac{1}{2} \).
- The intercept from \( x + 2 \) gives \( x = -2 \).
- The intercept from \( (8x - 5)^2 \) gives \( x = \frac{5}{8} \) (counted only once).

Each of these conditions gives us unique x-intercepts:

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

Thus, the function \( f(x) = 3(2x - 1)(x + 2)(8x - 5)^2 \) has **3 x-intercepts** when graphed.