Find all the zeros of the following function.
g(x) = x^3 + 10x^2 + 28x + 49
To find the zeros of the function g(x), we set g(x) equal to zero:
x^3 + 10x^2 + 28x + 49 = 0
To solve this equation, we can try to factor it or use the Rational Root Theorem.
Since the leading coefficient is 1, the possible rational zeros of the function are factors of the constant term, which is 49. The factors of 49 are ±1, ±7, and ±49.
Trying these values in the equation, we find that none of them are roots. Therefore, there are no rational zeros for g(x).
To find the zeros, we can use a graphing calculator or a numerical method such as the Newton-Raphson method.
Using a graphing calculator, we can find that g(x) has one real zero approximately at x = -4.277. The other two zeros are complex conjugates.
Therefore, the zeros of the function g(x) are approximately x = -4.277, x ≈ -2.861 + 1.167i, and x ≈ -2.861 - 1.167i.