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 x-intercepts of the function \( f(x) = 3(2x - 1)(x + 2)(8x - 5)^2 \), we need to determine when the function equals zero. The x-intercepts occur where \( f(x) = 0 \).

The function can be set equal to zero:

\[
3(2x - 1)(x + 2)(8x - 5)^2 = 0
\]

Since \( 3 \) is a constant and does not affect the roots, we can ignore it to find the x-intercepts:

\[
(2x - 1)(x + 2)(8x - 5)^2 = 0
\]

Next, we find the individual factors that equal zero:

1. \( 2x - 1 = 0 \)

Solving for \( x \):

\[
2x = 1 \quad \Rightarrow \quad x = \frac{1}{2}
\]

2. \( x + 2 = 0 \)

Solving for \( x \):

\[
x = -2
\]

3. \( (8x - 5)^2 = 0 \)

Solving for \( x \):

\[
8x - 5 = 0 \quad \Rightarrow \quad 8x = 5 \quad \Rightarrow \quad x = \frac{5}{8}
\]

Now, we take note of the multiplicity. The factor \( (8x - 5)^2 \) indicates that the root \( x = \frac{5}{8} \) has a multiplicity of 2, but it still counts as one x-intercept.

So, the unique x-intercepts are:

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

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