Complex Conjugates

Definition

Complex conjugates are a pair of complex numbers that have the same real part, but their imaginary parts have opposite signs. This relationship between complex numbers is important in the study of polynomial functions and partial fractions.

5 Must Know Facts For Your Next Test

1. The complex conjugate of $a + bi$ is $a - bi$, where $a$ is the real part and $b$ is the imaginary part.
2. When a polynomial function has complex roots, they always occur in pairs of complex conjugates.
3. Complex conjugates have the same absolute value, but their arguments (angles) differ by $180$ degrees.
4. The product of a complex number and its complex conjugate is always a real number.
5. Complex conjugates play a crucial role in simplifying expressions involving partial fractions with complex roots.

Review Questions

• Explain how complex conjugates are related to the roots of polynomial functions.
  • When a polynomial function has complex roots, they always occur in pairs of complex conjugates. This means that if $a + bi$ is a root of the polynomial, then $a - bi$ is also a root. The presence of complex conjugate roots in a polynomial function indicates that the function has no real roots, and the complex roots must be dealt with using specialized techniques, such as factoring or the use of the quadratic formula.
• Describe the relationship between the product of a complex number and its complex conjugate.
  • The product of a complex number $a + bi$ and its complex conjugate $a - bi$ is always a real number. This is because the imaginary parts of the two complex numbers cancel out, leaving only the real part: $(a + bi)(a - bi) = a^2 + b^2$. This property of complex conjugates is useful in simplifying expressions and solving problems involving partial fractions with complex roots.
• Analyze the role of complex conjugates in the context of partial fractions.
  • When dealing with partial fractions involving complex roots, the use of complex conjugates is essential. If the denominator of a partial fraction contains a quadratic expression with complex roots, those roots will appear as a pair of complex conjugates. By factoring the denominator and using the properties of complex conjugates, the partial fraction can be simplified and expressed in a more manageable form. This simplification process is a key step in the method of partial fractions, which is used to integrate rational functions.

Related terms

Complex Number: A complex number is a number that can be expressed in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Imaginary Unit: The imaginary unit, denoted as $i$, is defined as the square root of $-1$, meaning $i^2 = -1$.

Polynomial Function: A polynomial function is a function that can be expressed as the sum of a finite number of non-zero terms, each of which is a product of a constant and one or more variables raised to a non-negative integer power.