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