Problem 1 - Precalculus

Question

According to the fundamental theorem of algebra, which of the equations in Exercises 1 and 2 have at least one root? (a) \(x^{5}-14 x^{4}+8 x+53=0\) (b) \(4.17 x^{3}+2.06 x^{2}+0.01 x+1.23=0\) (c) \(i x^{2}+(2+3 i) x-17=0\) (d) \(x^{21}+3 x^{0.3}+1=0\)

Step-by-Step Solution

Answer

All four equations have at least one root.

Step 1: Understand the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-zero polynomial equation of degree \( n \) has exactly \( n \) roots in the complex number system (when counted with multiplicities). This implies that any polynomial equation of degree at least 1 has at least one root.

Step 2: Determine the Degree of Each Polynomial

- For (a) \( x^{5}-14 x^{4}+8 x+53=0 \), the degree is 5.- For (b) \( 4.17 x^{3}+2.06 x^{2}+0.01 x+1.23=0 \), the degree is 3.- For (c) \( i x^{2}+(2+3 i) x-17=0 \), the degree is 2.- For (d) \( x^{21}+3 x^{0.3}+1=0 \), the degree is 21.

Step 3: Apply the Fundamental Theorem

Since the Fundamental Theorem of Algebra applies to all polynomial equations of degree 1 or more, each of these equations, having degrees of 5, 3, 2, and 21 respectively, are guaranteed to have at least one root.

Key Concepts

Polynomial Degree

The degree of a polynomial is a critical concept in understanding many algebraic properties. It refers to the highest power of the variable present in the polynomial. For instance, in a polynomial like \(x^5 - 14x^4 + 8x + 53\), the term with the highest exponent, \(x^5\), determines the degree. Here, the degree is 5.
Polynomials are classified based on their degree:

• Linear polynomials have a degree of 1.
• Quadratic polynomials have a degree of 2.
• Cubic polynomials are of degree 3.
• Higher degree polynomials follow the same pattern (quartic, quintic, etc.).

Understanding the degree helps us apply the Fundamental Theorem of Algebra, which indicates that a polynomial of degree \( n \) will have exactly \( n \) roots in the complex number system. This includes all roots, real and complex, as well as their multiplicities.

Complex Roots

Complex roots can be a bit perplexing at first. They are numbers in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, equal to the square root of -1. Complex numbers are essential because they ensure that every polynomial equation can be solved.
The interesting part of complex roots is that they often come in conjugate pairs. If a polynomial with real coefficients has a complex root \(a + bi\), it will also have the root \(a - bi\). For example, a quadratic polynomial like \(ix^2 + (2 + 3i)x - 17 = 0\) may have complex solutions due to the presence of the imaginary unit \(i\).
Understanding complex roots is crucial when dealing with non-real solutions, as they allow polynomials to have the complete range of solutions as dictated by their degrees.

Equation Roots

The roots of an equation are the solutions that satisfy the equation when substituted in place of the variable. According to the Fundamental Theorem of Algebra, a non-zero polynomial equation of degree \( n \) will have \( n \) roots when counted with multiplicities and including complex numbers. This guarantees that for polynomials equations like those given in the exercise, each will have at least one root.
Here's how you can think about roots:

• Real roots are found where the polynomial crosses or touches the x-axis.
• Complex roots, which occur in conjugate pairs, do not correspond to a point on the traditional graph but are crucial in providing a full set of solutions.
• Repeated roots occur when the polynomial touches the x-axis but doesn't cross it, indicating a multiplicity greater than one.

Being able to identify the type and number of roots helps solve polynomials efficiently, whether through algebraic methods or modern computational tools.