Questions: The polynomial (f(x)) given below has -1 as a zero [f(x)=e^3+3 e^2+19 x+17] Find the other zeros of (f(x)). List the zeros separated by a comma.

Solution Steps

To find the zeros of the polynomial \( f(x) = x^3 + 3x^2 + 19x + 17 \), we know that -1 is a zero. We can use polynomial division to divide \( f(x) \) by \( x + 1 \) to find the quotient, which will be a quadratic polynomial. Then, we can solve the quadratic equation to find the other zeros.

The polynomial given is

\[ f(x) = x^3 + 3x^2 + 19x + 17 \]

We know that \( -1 \) is a zero of this polynomial.

Using polynomial division, we divide \( f(x) \) by \( x + 1 \) to find the quotient, which is a quadratic polynomial. The roots of this quadratic polynomial will give us the other zeros of \( f(x) \).

The roots of the polynomial \( f(x) \) are found to be

\[ -1 + 4j, \quad -1 - 4j, \quad -1 \]

where \( j \) is the imaginary unit. The zeros can be expressed as:

\[ -1 + 4i, \quad -1 - 4i \]

Final Answer