Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^5-6x^4+14x^3-20x^2+24x-16

Verified step by step guidance

1. Step 1: Use the Rational Root Theorem to identify possible rational roots. The possible rational roots are the factors of the constant term (16) divided by the factors of the leading coefficient (1).
2. Step 2: Test the possible rational roots using synthetic division or direct substitution to find any actual roots.
3. Step 3: Once a root is found, use synthetic division to divide the polynomial by the corresponding factor (e.g., if x = r is a root, divide by x - r) to reduce the polynomial's degree.
4. Step 4: Repeat the process of finding roots and dividing the polynomial until you have a quadratic or cubic polynomial.
5. Step 5: Use the quadratic formula or factorization to find the remaining complex roots of the reduced polynomial.

Key Concepts

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 is determined by the highest power of the variable. Understanding the structure of polynomial functions is essential for analyzing their behavior, including finding their zeros, which are the values of x that make the function equal to zero.

Complex Zeros

Complex zeros are solutions to polynomial equations that may not be real numbers. They can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The Fundamental Theorem of Algebra states that every polynomial of degree n has exactly n roots in the complex number system, which includes both real and non-real solutions.

Factoring and Synthetic Division

Factoring is the process of breaking down a polynomial into simpler components (factors) that can be multiplied to obtain the original polynomial. Synthetic division is a simplified form of polynomial long division that is particularly useful for dividing polynomials by linear factors. Both techniques are crucial for finding the zeros of a polynomial, as they help identify the roots more efficiently.