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

Verified step by step guidance

1. insert step 1: Recognize that the polynomial $f(x)=x^4+2x^2+1$ can be rewritten in a quadratic form by substituting $u=x^2$. This gives us $f(u)=u^2+2u+1$.
2. insert step 2: Notice that $f(u)=u^2+2u+1$ is a perfect square trinomial. It can be factored as $(u+1{)}^2$.
3. insert step 3: Substitute back $u=x^2$ into the factored form to get $(x^2+1{)}^2$.
4. insert step 4: Set $x^2+1=0$ to find the zeros. Solve for $x$ by isolating $x^2$ to get $x^2=-1$.
5. insert step 5: Solve $x^2=-1$ to find the complex zeros. The solutions are $x=i$ and $x=-i$, where $i$ is the imaginary unit.

Key Concepts

Complex Zeros

Complex zeros are the solutions to a polynomial equation that may include imaginary numbers. They occur when the polynomial does not intersect the x-axis in the real number system. For example, a polynomial of degree n can have up to n complex zeros, which may be real or non-real. Understanding how to find these zeros is crucial for analyzing polynomial functions.

Factoring Polynomials

Factoring polynomials involves expressing the polynomial as a product of simpler polynomials or linear factors. This process is essential for finding the zeros of the polynomial, as the zeros correspond to the values of x that make each factor equal to zero. Techniques such as grouping, using the quadratic formula, or recognizing special patterns (like perfect squares) are commonly employed in this process.

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial function of degree n has exactly n complex roots, counting multiplicities. This theorem guarantees that we can find all zeros of a polynomial, whether they are real or complex. It is a foundational concept in algebra that helps in understanding the behavior of polynomial functions and their graphs.