Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=5x³-9x²+28x+6

Verified step by step guidance

1. Start by writing down the polynomial function: \(f(x) = 5x^3 - 9x^2 + 28x + 6\).
2. Use the Rational Root Theorem to list all possible rational roots. These are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (6) and \(q\) divides the leading coefficient (5). So possible roots are \(\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{3}{5}, \pm \frac{6}{5}\).
3. Test these possible roots by substituting them into \(f(x)\) to find at least one root that makes \(f(x) = 0\). This root will be a zero of the polynomial.
4. Once a root \(r\) is found, perform polynomial division (either synthetic or long division) to divide \(f(x)\) by \((x - r)\), reducing the cubic polynomial to a quadratic polynomial.
5. Solve the resulting quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the remaining zeros, which may be real or complex.

Key Concepts

Complex Zeros of Polynomial Functions

Complex zeros are the values of x, possibly including imaginary numbers, that make the polynomial equal to zero. Finding all complex zeros involves solving the polynomial equation f(x) = 0, which may require factoring, synthetic division, or applying the quadratic formula when necessary.

Polynomial Division and Factoring

Polynomial division, including synthetic division, helps simplify higher-degree polynomials by dividing out known factors. Factoring breaks the polynomial into products of lower-degree polynomials, making it easier to find zeros by setting each factor equal to zero.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, is used to find exact roots of quadratic equations. When a polynomial is factored into a quadratic factor, this formula helps find complex or real zeros, especially when factoring is not straightforward.