Complex roots refer to the solutions to polynomial equations that have imaginary or non-real components. These roots arise when the discriminant of the equation is negative, indicating that the equation has no real solutions and instead has a pair of complex conjugate roots.
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Complex roots always occur in pairs of complex conjugates, meaning that if $z = a + bi$ is a root, then $\overline{z} = a - bi$ is also a root.
The Fundamental Theorem of Algebra states that every non-constant polynomial equation has at least one complex root.
In the polar form of a complex number, the complex roots can be expressed as $r(\cos(\theta) \pm i\sin(\theta))$, where $r$ is the modulus and $\theta$ is the argument.
When working with partial fractions, complex roots lead to the introduction of conjugate pairs of linear factors in the denominator of the fraction.
The presence of complex roots in a polynomial equation indicates that the equation has no real solutions and that the solutions must be found using complex number arithmetic.
Review Questions
Explain how the sign of the discriminant of a quadratic equation relates to the nature of its roots.
The sign of the discriminant of a quadratic equation $ax^2 + bx + c = 0$ determines the nature of the roots. If the discriminant $b^2 - 4ac$ is positive, the equation has two real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex conjugate roots. This is because a negative discriminant indicates that the equation has no real solutions, and the roots must be found using complex number arithmetic.
Describe the relationship between complex roots and the polar form of complex numbers.
When dealing with complex roots, the polar form of complex numbers can be a useful representation. In the polar form, a complex number $z = a + bi$ can be expressed as $z = r(\cos(\theta) + i\sin(\theta))$, where $r$ is the modulus (or magnitude) and $\theta$ is the argument (or angle). For complex roots, the roots can be expressed in this polar form as $r(\cos(\theta) \pm i\sin(\theta))$, where the $\pm$ sign indicates the complex conjugate pair of roots.
Explain how complex roots impact the process of finding partial fractions.
The presence of complex roots in the denominator of a rational function affects the process of finding the partial fraction decomposition. When the denominator has a quadratic factor with complex roots, the partial fraction decomposition will include conjugate pairs of linear factors, rather than distinct real linear factors. This requires the use of complex number arithmetic and the introduction of complex-valued coefficients in the partial fraction expansion. The complex roots and their conjugates must be accounted for in order to properly decompose the rational function into its partial fractions.
Related terms
Polynomial Equation: An equation that can be written in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0$, where $a_n, a_{n-1}, ..., a_1, a_0$ are real numbers and $n$ is a positive integer.
The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is the expression $b^2 - 4ac$. The sign of the discriminant determines the nature of the roots of the equation.
If a complex number is of the form $a + bi$, then its complex conjugate is $a - bi$. Complex conjugates have the same real part but opposite imaginary parts.