How many x-intercepts does the function f(x)=3(2x−1)(x+2)(8x−5)^2
have when graphed?
The x-intercepts of a function are the points where the graph intersects the x-axis. In order for the function to have an x-intercept, the value of f(x) must equal 0.
To find the x-intercepts of the function f(x) = 3(2x−1)(x+2)(8x−5)^2, set f(x) equal to 0:
0 = 3(2x−1)(x+2)(8x−5)^2
So, the function will have an x-intercept when 3(2x−1)(x+2)(8x−5)^2 = 0.
The function has 3 factors: (2x−1), (x+2), and (8x−5)^2.
The x-intercepts are determined by setting each factor equal to zero:
(2x−1) = 0 → 2x = 1 → x = 1/2
(x+2) = 0 → x = -2
(8x−5)^2 = 0 → 8x−5 = 0 → 8x = 5 → x = 5/8
Thus, the function has 3 x-intercepts: x = 1/2, x = -2, and x = 5/8.
To find the x-intercepts of a function, we set the value of the function equal to zero and solve for x.
f(x) = 3(2x−1)(x+2)(8x−5)^2
Setting f(x) equal to zero:
0 = 3(2x−1)(x+2)(8x−5)^2
Now, we can identify that the x-intercepts occur when any of the factors (2x−1), (x+2), or (8x−5)^2 equal zero.
Setting each factor equal to zero and solving for x:
1) 2x−1 = 0
2x = 1
x = 1/2
2) x+2 = 0
x = -2
3) (8x−5)^2 = 0
Taking the square root of both sides:
8x−5 = 0
8x = 5
x = 5/8
Therefore, the function f(x) has three x-intercepts at x = 1/2, x = -2, and x = 5/8.
To find the x-intercepts of a function, you need to determine the values of x where the function equals zero. In other words, you're looking for the values of x that make f(x) = 0.
In this case, the function f(x) = 3(2x−1)(x+2)(8x−5)^2.
To find the x-intercepts, you can set f(x) equal to zero and solve for x:
0 = 3(2x−1)(x+2)(8x−5)^2
Now, since the function is multiplied by three terms: (2x−1)(x+2)(8x−5)^2, you can use the zero product property which states that if the product of several factors equals zero, then at least one of the factors must be zero.
So, you can set each factor equal to zero:
2x−1 = 0 (Factor 1)
x+2 = 0 (Factor 2)
(8x−5)^2 = 0 (Factor 3)
Solving each equation separately:
From Factor 1:
2x−1 = 0
Add 1 to both sides:
2x = 1
Divide both sides by 2:
x = 1/2
From Factor 2:
x+2 = 0
Subtract 2 from both sides:
x = -2
From Factor 3:
(8x−5)^2 = 0
Take the square root of both sides:
8x−5 = 0
Add 5 to both sides:
8x = 5
Divide both sides by 8:
x = 5/8
So, we have found three values of x that make f(x) = 0: x = 1/2, x = -2, and x = 5/8.
Therefore, the function f(x) = 3(2x−1)(x+2)(8x−5)^2 will have three x-intercepts when graphed.