Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴+2x²+1

Verified step by step guidance

1. Recognize that the polynomial ƒ(x) = x^4 + 2x^2 + 1 is a quartic polynomial, but it can be treated as a quadratic in terms of x^2. To do this, let \( y = x^2 \). Then rewrite the polynomial as \( y^2 + 2y + 1 \).
2. Notice that the quadratic in \( y \) is \( y^2 + 2y + 1 \), which can be factored or recognized as a perfect square trinomial. Factor it as \( (y + 1)^2 \).
3. Set the factored form equal to zero to find the zeros in terms of \( y \): \( (y + 1)^2 = 0 \) which implies \( y + 1 = 0 \). Solve for \( y \) to get \( y = -1 \).
4. Recall that \( y = x^2 \), so substitute back to get \( x^2 = -1 \). To find \( x \), take the square root of both sides, remembering to include both positive and negative roots: \( x = \pm \sqrt{-1} \).
5. Since \( \sqrt{-1} = i \) (the imaginary unit), the complex zeros are \( x = i \) and \( x = -i \). Because the factor was squared, each zero has multiplicity 2, so list each zero twice.

Key Concepts

Complex Zeros of Polynomial Functions

Complex zeros are values of x, possibly including imaginary numbers, that make the polynomial equal to zero. Finding all complex zeros involves solving the polynomial equation, which may require factoring or using formulas, and includes real and non-real solutions.

Factoring Polynomials

Factoring breaks down a polynomial into simpler polynomials whose product equals the original. For quartic polynomials like x⁴ + 2x² + 1, recognizing patterns such as quadratic forms or perfect squares helps simplify the problem and find zeros more easily.

Quadratic Substitution Method

This method involves substituting a variable (e.g., y = x²) to transform a higher-degree polynomial into a quadratic form. Solving the quadratic in y and then back-substituting helps find the original variable's zeros, including complex ones.