Question
How many x-intercepts would the function shown below have?
f(x)=(x^2-2)(6x+5)(x^2+4)
f(x)=(x^2-2)(6x+5)(x^2+4)
Answers
Answered by
GPT 3.5
To find the x-intercepts of a function, we need to find the values of x for which f(x) = 0. Therefore, we need to solve the equation: (x^2-2)(6x+5)(x^2+4) = 0.
This equation will be satisfied if any of the three factors (x^2-2), (6x+5), and (x^2+4) are equal to zero.
Solving the equation x^2-2 = 0, we find x = ±√2, which gives us two x-intercepts.
Solving the equation 6x+5 = 0, we find x = -5/6, which gives us one x-intercept.
Solving the equation x^2+4 = 0, we find no real solutions, as it is a quadratic equation with no real roots. However, it has two complex roots, so we can also say that it has no real x-intercepts.
Therefore, the function f(x) = (x^2-2)(6x+5)(x^2+4) has a total of 2 + 1 + 0 = 3 x-intercepts.
This equation will be satisfied if any of the three factors (x^2-2), (6x+5), and (x^2+4) are equal to zero.
Solving the equation x^2-2 = 0, we find x = ±√2, which gives us two x-intercepts.
Solving the equation 6x+5 = 0, we find x = -5/6, which gives us one x-intercept.
Solving the equation x^2+4 = 0, we find no real solutions, as it is a quadratic equation with no real roots. However, it has two complex roots, so we can also say that it has no real x-intercepts.
Therefore, the function f(x) = (x^2-2)(6x+5)(x^2+4) has a total of 2 + 1 + 0 = 3 x-intercepts.