Find the zeros (if any) of the rational function. (If there are no zeros, enter NONE.)
g(x) = (x^3-64)/(x^2+2)
need help solving
A fraction can only be zero if its numerator is zero
so
x^3 - 64 = 0
difference of cube factoring ....
(x-4)(x^2 + 4x + 16) = 0
x = 4 or x = (-4 ± √(16- 64) )/2
x = 4 or x = -2 ± 4√3 i
so the graph should cross the x-axis only once, at x = 4
confirmation:
http://www.wolframalpha.com/input/?i=plot+y+%3D+%28x%5E3-64%29%2F%28x%5E2%2B2%29+for+x+%3D++-20+to+20
To find the zeros of the rational function g(x) = (x^3-64)/(x^2+2), we need to find the values of x that make the numerator equal to zero.
The numerator is x^3 - 64, which we can factor using the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
In this case, a = x and b = 4, since 4 cubed is 64. So we can factor the numerator as (x - 4)(x^2 + 4x + 16).
Setting the numerator equal to zero, we have:
(x - 4)(x^2 + 4x + 16) = 0.
For the numerator to equal zero, either (x - 4) must be zero or (x^2 + 4x + 16) must be zero.
If x - 4 = 0, then x = 4.
To solve x^2 + 4x + 16 = 0, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a),
where a, b, and c are the coefficients of the quadratic equation. In this case, a = 1, b = 4, and c = 16.
Plugging these values into the quadratic formula, we have:
x = (-4 ± sqrt(4^2 - 4(1)(16))) / (2(1)),
x = (-4 ± sqrt(16 - 64)) / 2,
x = (-4 ± sqrt(-48)) / 2,
x = (-4 ± 4i√3) / 2,
x = -2 ± 2i√3.
Therefore, the zeros of the rational function g(x) = (x^3-64)/(x^2+2) are x = 4, x = -2 + 2i√3, and x = -2 - 2i√3.