To determine the number of x-intercepts for the function \( f(x) = 3(2x - 1)(x + 2)(8x - 5)^2 \), we need to find the values of \( x \) for which \( f(x) = 0 \).
The function \( f(x) \) can be set to zero to identify the x-intercepts:
\[ 3(2x - 1)(x + 2)(8x - 5)^2 = 0 \]
Since 3 is a non-zero constant, we can ignore it for finding x-intercepts, so we only need to solve:
\[ (2x - 1)(x + 2)(8x - 5)^2 = 0 \]
Now, we solve each factor separately:
-
\( 2x - 1 = 0 \) \[ 2x = 1 \implies x = \frac{1}{2} \] This gives us one x-intercept at \( x = \frac{1}{2} \).
-
\( x + 2 = 0 \) \[ x = -2 \] This gives us another x-intercept at \( x = -2 \).
-
\( (8x - 5)^2 = 0 \) \[ 8x - 5 = 0 \implies 8x = 5 \implies x = \frac{5}{8} \] The factor \( (8x - 5)^2 \) indicates that this x-intercept occurs with multiplicity 2.
Now, let’s summarize the x-intercepts we found:
- \( x = \frac{1}{2} \) (multiplicity 1)
- \( x = -2 \) (multiplicity 1)
- \( x = \frac{5}{8} \) (multiplicity 2)
However, the multiplicity does not affect the count of distinct x-intercepts.
So, we have three distinct x-intercepts:
- \( x = \frac{1}{2} \)
- \( x = -2 \)
- \( x = \frac{5}{8} \)
Thus, the total number of x-intercepts for the function \( f(x) \) is 3.
1 answer