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