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 \( x \) for which \( f(x) = 0 \).
The function can be set equal to zero as follows:
\[ 3(2x - 1)(x + 2)(8x - 5)^2 = 0 \]
Since the factor of 3 is a constant and does not affect the roots, we can focus on the polynomial part:
\[ (2x - 1)(x + 2)(8x - 5)^2 = 0 \]
Now we will identify the roots of each factor:
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From \( 2x - 1 = 0 \): \[ 2x = 1 \implies x = \frac{1}{2} \]
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From \( x + 2 = 0 \): \[ x = -2 \]
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From \( (8x - 5)^2 = 0 \): \[ 8x - 5 = 0 \implies 8x = 5 \implies x = \frac{5}{8} \] The factor \( (8x - 5)^2 \) gives a double root at \( x = \frac{5}{8} \).
Now we summarize the roots:
- \( x = \frac{1}{2} \) (single root)
- \( x = -2 \) (single root)
- \( x = \frac{5}{8} \) (double root)
X-intercepts correspond to the unique values of \( x \) where the function is equal to zero. The roots we have found are:
- \( x = \frac{1}{2} \)
- \( x = -2 \)
- \( x = \frac{5}{8} \)
Counting the unique roots, we have three distinct x-intercepts:
- \( \frac{1}{2} \)
- \( -2 \)
- \( \frac{5}{8} \)
Thus, the function \( f(x) \) has 3 x-intercepts.
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