Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴+4x³+6x²+4x+1

Verified step by step guidance

1. Recognize that the polynomial ƒ(x) = x^4 + 4x^3 + 6x^2 + 4x + 1 resembles the expansion of a binomial expression. Recall the binomial theorem and check if it matches the expansion of (x + 1)^n for some n.
2. Write down the binomial expansion for (x + 1)^4, which is given by the formula: \[(x + 1)^4 = \sum_{k=0}^4 \binom{4}{k} x^{4-k} 1^k = x^4 + 4x^3 + 6x^2 + 4x + 1.\] Confirm that this matches the given polynomial exactly.
3. Since the polynomial is equal to \((x + 1)^4\), set the equation to zero to find the zeros: \[(x + 1)^4 = 0.\]
4. Solve the equation by taking the fourth root of both sides, which leads to \(x + 1 = 0\) because the only root of zero raised to any power is zero itself.
5. Find the zero by isolating \(x\): \[x = -1.\] Since the factor is raised to the fourth power, the zero \(x = -1\) has multiplicity 4, meaning it is a repeated root four times.

Key Concepts

Polynomial Zeros and Roots

Polynomial zeros are the values of x for which the polynomial equals zero. Finding these roots involves solving the equation f(x) = 0. Zeros can be real or complex numbers, and their multiplicity indicates how many times a root repeats.

Factoring Polynomials

Factoring breaks down a polynomial into simpler polynomials whose product equals the original. Recognizing patterns like the binomial expansion or perfect powers helps simplify the polynomial, making it easier to find zeros by setting each factor to zero.

Complex Numbers and Roots

Complex numbers include real and imaginary parts and are essential when polynomial roots are not real. Understanding how to express and manipulate complex roots, including using the Fundamental Theorem of Algebra, allows finding all zeros of a polynomial.