Question

How many x-intercepts does the function f(x)=3(2x−1)(x+2)(8x−5)2
have 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 the values of \( x \) for which \( f(x) = 0 \).

The function is a product of several factors, and it will equal zero whenever any of the factors is zero. Therefore, we will set each factor to zero and solve for \( x \):

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

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

3. **For the factor \( (8x - 5)^2 = 0 \)**:
\[
8x - 5 = 0 \implies 8x = 5 \implies x = \frac{5}{8}
\]
The factor \( (8x - 5)^2 \) is a square, which means it touches the x-axis at this point but does not cross it. Thus, it provides only one x-intercept, even though it represents a double root.

Now, we summarize the x-intercepts:

- From \( 2x - 1 = 0 \): \( x = \frac{1}{2} \) (one x-intercept)
- From \( x + 2 = 0 \): \( x = -2 \) (one x-intercept)
- From \( (8x - 5)^2 = 0 \): \( x = \frac{5}{8} \) (one x-intercept that does not cross the axis)

Thus, the function \( f(x) \) has three x-intercepts:

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

Putting this together, the total number of x-intercepts when we consider distinct solutions is **3**.

The function has **three x-intercepts**.