Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=2x^4-x^3+7x^2-4x-4

Verified step by step guidance

1. insert step 1: Start by identifying the polynomial function, which is \( f(x) = 2x^4 - x^3 + 7x^2 - 4x - 4 \).
2. insert step 2: Use the Rational Root Theorem to list possible rational roots. These are the factors of the constant term (-4) divided by the factors of the leading coefficient (2).
3. insert step 3: Test the possible rational roots using synthetic division or direct substitution to find any actual roots.
4. insert step 4: Once a root is found, use polynomial division to divide the polynomial by \( x - \text{(root)} \) to reduce the polynomial's degree.
5. insert step 5: Repeat the process for the reduced polynomial to find all roots, including complex ones, using methods such as factoring, completing the square, or the quadratic formula if necessary.

Key Concepts

Complex Zeros

Complex zeros are solutions to polynomial equations that may include real and imaginary numbers. A complex zero can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. Understanding complex zeros is essential for analyzing polynomial functions, especially when the polynomial does not have real roots.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The degree of the polynomial, determined by the highest power of the variable, influences the number of zeros it can have. For example, a fourth-degree polynomial can have up to four complex zeros, including real zeros.

Finding Zeros

Finding the zeros of a polynomial involves determining the values of the variable that make the polynomial equal to zero. This can be achieved through various methods, including factoring, synthetic division, or applying the Rational Root Theorem. For polynomials with complex coefficients, techniques such as the quadratic formula or numerical methods may be necessary to identify all zeros accurately.