How many x-intercepts does the function f(x)=3(2x−1)(x+2)(8x−5)^2 have when graphed? (1 point)
The function f(x) has blank x-intercepts when graphed.
The function f(x)=3(2x−1)(x+2)(8x−5)^2 has two x-intercepts when graphed.
To find the x-intercepts of a function, we need to solve for the values of x when f(x) = 0.
Given the function f(x) = 3(2x−1)(x+2)(8x−5)^2, we can set f(x) equal to zero and solve for x.
0 = 3(2x−1)(x+2)(8x−5)^2
Since the function is multiplied by three factors, we know that the graph will intersect the x-axis at the x-values where each factor is equal to zero.
Setting each factor equal to zero, we have:
2x−1 = 0 --> 2x = 1 --> x = 1/2
x+2 = 0 --> x = -2
8x−5 = 0 --> 8x = 5 --> x = 5/8
Therefore, the function f(x) = 3(2x−1)(x+2)(8x−5)^2 has three x-intercepts when graphed.
The x-intercepts are x = 1/2, x = -2, and x = 5/8.
To determine the number of x-intercepts of a function, we need to examine the factors of the function and determine when these factors equal zero.
In this case, the function f(x) = 3(2x−1)(x+2)(8x−5)^2 has three factors: (2x−1), (x+2), and (8x−5)^2.
1. Setting (2x−1) equal to zero:
2x − 1 = 0
Solving this equation, we get:
2x = 1
x = 1/2
2. Setting (x+2) equal to zero:
x + 2 = 0
Solving this equation, we get:
x = -2
3. Setting (8x−5)^2 equal to zero:
(8x−5)^2 = 0
Taking the square root of both sides, we get:
8x − 5 = 0
Solving this equation, we get:
8x = 5
x = 5/8
So, the function f(x) has three x-intercepts at x = 1/2, x = -2, and x = 5/8 when graphed.