Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^4+4x^3+6x^2+4x+1

Verified step by step guidance

1. insert step 1: Recognize that the polynomial \( f(x) = x^4 + 4x^3 + 6x^2 + 4x + 1 \) is a binomial expansion.
2. insert step 2: Notice that the polynomial resembles the expansion of \((x+1)^4\).
3. insert step 3: Verify by expanding \((x+1)^4\) to ensure it matches \( f(x) \).
4. insert step 4: Since \( f(x) = (x+1)^4 \), the zeros of the polynomial are the solutions to \((x+1)^4 = 0\).
5. insert step 5: Solve \((x+1)^4 = 0\) to find the complex zeros, considering multiplicity.

Key Concepts

Complex Zeros

Complex zeros are solutions to polynomial equations that may include imaginary numbers. They occur when the polynomial does not intersect the x-axis at certain points, leading to roots of the form a + bi, where a and b are real numbers. Understanding how to find complex zeros is essential for analyzing polynomial functions, especially when they have even degrees or repeated 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 and type of zeros it can have. For example, a fourth-degree polynomial can have up to four zeros, which may be real or complex.

Factoring and the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial function has at least one complex root, and the total number of roots (counting multiplicities) equals the degree of the polynomial. Factoring polynomials can help identify these roots, as it breaks down the polynomial into simpler components, making it easier to find both real and complex zeros.