Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $\text{ƒ}\left(x\right)=2x^4-x^3+7x^2-4x-4$

Verified step by step guidance

1. Start by writing down the polynomial function: \(f(x) = 2x^4 - x^3 + 7x^2 - 4x - 4\).
2. Attempt to find rational zeros using the Rational Root Theorem, which suggests possible roots of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (-4) and \(q\) divides the leading coefficient (2). List these possible rational roots.
3. Test each possible rational root by substituting into \(f(x)\) or by using synthetic division to check if it yields zero. When a root is found, factor it out from the polynomial.
4. After factoring out the found root(s), reduce the polynomial to a lower degree and repeat the process to find other roots. If the remaining polynomial is quadratic, use the quadratic formula to find the complex zeros.
5. Express all zeros found, including any complex zeros, in exact form. Remember that complex zeros often come in conjugate pairs if the polynomial has real coefficients.

Key Concepts

Complex Zeros of Polynomial Functions

Complex zeros are the values of x, including real and non-real complex numbers, that make the polynomial equal to zero. According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n complex zeros, counting multiplicities. Finding these zeros involves solving the polynomial equation ƒ(x) = 0.

Polynomial Division and Factoring

To find zeros of higher-degree polynomials, techniques like synthetic division or long division help factor the polynomial into lower-degree polynomials. Factoring simplifies the problem by breaking the polynomial into products of linear or quadratic factors, which can then be solved individually for zeros.

Quadratic Formula and Solving Quadratics

When factoring leads to quadratic factors that cannot be factored further, the quadratic formula is used to find their zeros. The formula x = (-b ± √(b² - 4ac)) / 2a provides exact solutions, including complex ones when the discriminant is negative, ensuring all zeros of the polynomial are found.