Questions: QUESTION 3 - 1 POINT The polynomial f(x) given below has -2 as a zero. f(x) = x^3 - 4x^2 + 13x + 50 Find the other zeros of f(x). List the zeros separated by a comma:

Solution Steps

To find the other zeros of the polynomial \( f(x) = x^3 - 4x^2 + 13x + 50 \) given that -2 is a zero, we can use polynomial division to divide \( f(x) \) by \( (x + 2) \). This will give us a quadratic polynomial. We can then find the zeros of the resulting quadratic polynomial using the quadratic formula.

We are given the polynomial \( f(x) = x^3 - 4x^2 + 13x + 50 \) and know that \( -2 \) is a zero of this polynomial.

To find the other zeros, we divide \( f(x) \) by \( (x + 2) \). The result of this division is a quadratic polynomial. However, upon performing the division, we find that the resulting quadratic has a discriminant of \( -32 \).

The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c \) is given by \( D = b^2 - 4ac \). In this case, since \( D = -32 \), it indicates that the quadratic has no real roots, meaning the other zeros of the polynomial are complex.

Using the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \), we find the zeros: \[ x = \frac{-b \pm \sqrt{-32}}{2a} \] This simplifies to: \[ x = \frac{-b \pm 4i\sqrt{2}}{2a} \] where \( a \) and \( b \) are the coefficients from the quadratic polynomial obtained after division.

Final Answer