Final answer:
To find the other zeros of f(x)=x³+5x²-2x-10, we can use synthetic division to divide the polynomial by (x+5). The result is a quadratic equation, which can be solved using the quadratic formula. However, when we apply the quadratic formula, we find that the equation has no real solutions. Therefore, the only zero of f(x) is x=-5.
Step-by-step explanation:
The given polynomial function is f(x) = x³+5x²-2x-10. It is known that x=-5 is a zero of this function. To find the other zeros of f(x), we can use polynomial division or synthetic division. Let's use synthetic division:
Divide f(x) by (x+5):
-5 │ 1 5 -2 -10
-5 -10 60
---------------- 1 0 -12 50
The result of synthetic division is 1x² - 12x + 50. Now, we have a quadratic equation. To find the zeros of this equation, we can use the quadratic formula:
x = (-b ± √(b²-4ac)) / (2a)
For the quadratic equation 1x² - 12x + 50 = 0, a = 1, b = -12, and c = 50. Plugging these values into the quadratic formula, we get:
x = (-(-12) ± √((-12)² - 4(1)(50))) / (2(1))
Simplifying further:
x = (12 ± √(144 - 200)) / 2
x = (12 ± √(-56)) / 2
We have a negative value inside the square root, which means the quadratic equation has no real solutions. Therefore, the only zero of f(x) is x = -5.