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:
-
\( 2x - 1 = 0 \)
Solving for \( x \): \[ 2x = 1 \ x = \frac{1}{2} \] -
\( x + 2 = 0 \)
Solving for \( x \): \[ x = -2 \] -
\( (8x - 5)^2 = 0 \)
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:
- \( \frac{1}{2} \)
- \( -2 \)
- \( \frac{5}{8} \)
Thus, the function \( f(x) \) has 3 x-intercepts when graphed.
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