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 \( x \):
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For the factor \( 2x - 1 = 0 \): \[ 2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2} \]
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For the factor \( x + 2 = 0 \): \[ x + 2 = 0 \implies x = -2 \]
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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:
- \( x = \frac{1}{2} \)
- \( x = -2 \)
- \( x = \frac{5}{8} \) (double root)
Putting this together, the total number of x-intercepts when we consider distinct solutions is 3.
The function has three x-intercepts.