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# 6.12: Finding Imaginary Solutions

Difficulty Level: At Grade Created by: CK-12
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Practice Fundamental Theorem of Algebra
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### Vocabulary Language: English

Complex Conjugate

Complex conjugates are pairs of complex binomials. The complex conjugate of $a+bi$ is $a-bi$. When complex conjugates are multiplied, the result is a single real number.

complex number

A complex number is the sum of a real number and an imaginary number, written in the form $a + bi$.

conjugate pairs theorem

The conjugate pairs theorem states that if $f(z)$ is a polynomial of degree $n$, with $n\ne0$ and with real coefficients, and if $f(z_{0})=0$, where $z_{0}=a+bi$, then $f(z_{0}^{*})=0$. Where $z_{0}^{*}$ is the complex conjugate of $z_{0}$.

fundamental theorem of algebra

The fundamental theorem of algebra states that if $f(x)$ is a polynomial of degree $n\ge 1$, then $f(x)$ has at least one zero in the complex number domain. In other words, there is at least one complex number $c$ such that $f(c)=0$. The theorem can also be stated as follows: an $n^{th}$ degree polynomial with real or complex coefficients has, with multiplicity, exactly $n$ complex roots.

Imaginary Number

An imaginary number is a number that can be written as the product of a real number and $i$.

Imaginary Numbers

An imaginary number is a number that can be written as the product of a real number and $i$.

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.

Roots

The roots of a function are the values of x that make y equal to zero.

Zero

The zeros of a function $f(x)$ are the values of $x$ that cause $f(x)$ to be equal to zero.

Zeroes

The zeroes of a function $f(x)$ are the values of $x$ that cause $f(x)$ to be equal to zero.

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