Question

How many x-intercepts does the function f(x)=3(2x−1)(x+2)(8x−5)^2 have when graphed?

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the x-intercepts of a function, we need to find the values of x for which f(x) = 0.

In this case, the function is f(x) = 3(2x - 1)(x + 2)(8x - 5)^2.

Setting f(x) equal to 0, we have:
3(2x - 1)(x + 2)(8x - 5)^2 = 0.

Since the function is multiplied by three factors, we can set each factor equal to zero and solve for x.

The first factor, 2x - 1, equals zero when x = 1/2.

The second factor, x + 2, equals zero when x = -2.

The third factor, (8x - 5)^2, equals zero when 8x - 5 = 0. Solving for x, we find x = 5/8.

Therefore, the function has three x-intercepts at x = 1/2, x = -2, and x = 5/8.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the number of x-intercepts for the function f(x)=3(2x−1)(x+2)(8x−5)^2, you need to determine the number of real roots or solutions for the equation f(x) = 0. Since an x-intercept represents the point where the graph of the function crosses the x-axis, the x-values that satisfy f(x) = 0 will be the x-intercepts.

To find the x-intercepts, set f(x) equal to zero:

3(2x−1)(x+2)(8x−5)^2 = 0

Now, the equation will have x-intercepts when any of the three factors equal zero. So, check each factor separately and solve for x:

2x − 1 = 0 -> 2x = 1 -> x = 1/2

x + 2 = 0 -> x = -2

8x − 5 = 0 -> 8x = 5 -> x = 5/8

Since each factor can yield a unique x-intercept, there are three x-intercepts for the function f(x)=3(2x−1)(x+2)(8x−5)^2 when graphed.