Problem 6 - Algebra for College Students

Question

After reading this section, write out the answers to these questions. Use complete sentences. How many solutions are there to any quadratic equation in the complex number system?

Step-by-Step Solution

Answer

Any quadratic equation in the complex number system has 2 solutions.

Step 1: Understand Quadratic Equations

A quadratic equation is in the form of ax^2 + bx + c = 0 where a, b, and c are constants.

Step 2: Complex Number System

In the complex number system, numbers include real numbers and imaginary numbers (which are multiples of √(-1)).

Step 3: Fundamental Theorem Of Algebra

According to the Fundamental Theorem of Algebra, every non-zero polynomial equation has exactly as many roots as its degree when counted with multiplicity and including complex numbers.

Step 4: Degree of Quadratic Equation

A quadratic equation has a degree of 2, meaning it is a second-degree polynomial.

Step 5: Number of Solutions

As a quadratic equation is a second-degree polynomial, it will have exactly two solutions in the complex number system.

Key Concepts

complex number system

The complex number system expands upon the real number system by including numbers that involve the square root of negative one, denoted as \(i\). A complex number is generally written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Here are some key points to help understand complex numbers better:

• Real numbers are a subset of complex numbers where the imaginary part (\bi\term) is zero.
• Imaginary numbers are those where the real part (\(a\)) is zero, such as \(5i\) or \( -7i \).
• The arithmetic with complex numbers follows specific rules; for instance, \(i^2 = -1\).

This system allows us to solve polynomials that have no real solutions. For example, the equation \(x^2 + 1 = 0\) has no real solution, but it has two complex solutions: \(x = i \) and \(x = -i \). This capability is crucial in various fields of science and engineering.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a central principle in algebra, stating that every non-zero polynomial equation has exactly as many roots as its degree, when counted with multiplicity and including complex numbers.
This means that:

• If you have a quadratic equation, which is a polynomial of degree 2, it will have two solutions.
• Polynomials of degree 3 will have three solutions, and so on.

It's important to note that these roots may be real or complex. For example, the polynomial \(x^2 - 2x + 5 = 0\), has two complex roots: \(1 + 2i \) and \(1 - 2i \). Hence, the theorem ensures the existence of solutions in the broader complex number system, which cannot always be found within the real numbers alone.

degree of polynomial

The degree of a polynomial is determined by the highest power of the variable in the polynomial expression. For example, in the quadratic equation \(ax^2 + bx + c = 0\), the degree is 2 because the highest exponent of \(x\) is 2. Here are a few essential points on polynomial degrees:

• Degree 1: These are linear polynomials, such as \(ax + b = 0\), with exactly one solution.
• Degree 2: These are quadratic polynomials, like \(ax^2 + bx + c = 0\), which have exactly two solutions in the complex number system.
• Higher Degree: Polynomials with degree \(n\) will have \ roots. For example, \(x^3 + 2x + 1 = 0\) is a third-degree polynomial and has three roots.

Understanding the degree of a polynomial helps us predict the number of solutions we can expect, thanks to the Fundamental Theorem of Algebra.