Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=4x³+3x²+8x+6

Verified step by step guidance

1. Start by writing down the polynomial function: \(f(x) = 4x^3 + 3x^2 + 8x + 6\).
2. Use the Rational Root Theorem to list possible rational zeros. These are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (6) and \(q\) divides the leading coefficient (4). So possible rational roots are \(\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{1}{4}, \pm \frac{3}{4}\).
3. Test these possible roots by substituting them into \(f(x)\) to find any actual zeros. When you find a root \(r\), use polynomial division or synthetic division to divide \(f(x)\) by \((x - r)\) to reduce the cubic to a quadratic.
4. Once you have the quadratic factor from the division, solve it using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are coefficients from the quadratic.
5. Combine the rational root(s) found and the solutions from the quadratic formula to list all complex zeros of the polynomial, including any repeated roots if applicable.

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 reduced to a quadratic factor, this formula helps find real or complex zeros depending on the discriminant's value.