Problem 67 - Precalculus

Question

Determine whether the statement is true or false. There are always \(n\) distinct real solutions of the equation \(x^{n}-a=0,\) where \(a\) is not zero.

Step-by-Step Solution

Answer

False; it doesn't always have \(n\) distinct real solutions.

Step 1: Read the Statement

We must determine if the statement is true or false:
Determine whether the statement is true or false.
There are always \(n\) distinct real solutions of the equation \(x^{n}-a=0,\)
where \(a\) is not zero.

Step 2: Analyze the Statement

We examine the claim using relevant definitions, properties, or theorems.

Step 3: Conclusion

False; it doesn't always have \(n\) distinct real solutions.

Key Concepts

Real Solutions

When discussing polynomial equations such as \(x^n - a = 0\), understanding the nature of real solutions is crucial. Real solutions are the values of \(x\) which satisfy the equation and can be plotted on the real number line. In the context of this polynomial equation: - The equation represents a polynomial of degree \(n\), which implies it might have up to \(n\) real solutions. - However, these solutions aren’t always real; they can be complex numbers as well.Whether the solutions are real depends on certain factors, including the degree of the polynomial and the value of \(a\). For instance, if \(n\) is even and \(a > 0\), one might find only a limited number of real solutions, or even none, because complex roots appear in conjugate pairs.Being familiar with the behavior of higher-degree polynomials, including how they interact with the y-axis at least once, helps us anticipate when real solutions exist. Nevertheless, one cannot assume that all potential solutions are real unless evaluated specifically within the context of given parameters.

Distinct Solutions

Distinct solutions or distinct roots indicate that each solution is different from the others. For polynomial equations, having \(n\) distinct solutions implies that every solution \(x_1, x_2, ..., x_n\) is unique. Here are some significant points:- Identifying distinct solutions is dependent on the discriminant of the polynomial. For example, if the discriminant is non-zero, the roots are generally distinct.- Distinct real solutions occur when each real root is separate on the real number line, without repeating.Given the equation \(x^n - a = 0\), assuming all \(n\) solutions are distinct is incorrect unless additional specific information about the polynomial is provided. In fact, the assumption of distinct real solutions could fail under certain conditions, especially when dealing with repeated roots, where the polynomial graph touches or crosses the x-axis at the same point multiple times. This phenomenon is often associated with multiplicity greater than one, making it essential to examine the structure of the equation carefully.

Degree of Polynomial

The degree of a polynomial is a fundamental concept and refers to the highest power of the variable in the equation, which in this case is \(n\). It holds a significant role in determining the potential number of solutions.- The degree tells us the maximum number of solutions a polynomial can have. This includes real and complex solutions.- For the equation \(x^n - a = 0\), the degree is \(n\), meaning there can be up to \(n\) roots or solutions overall.However, while the degree informs us about the potential number of solutions, it does not guarantee \(n\) distinct real solutions. The Fundamental Theorem of Algebra ensures \(n\) total roots, but they may not all be distinct or even real. This is because the roots may include complex numbers, depending on specific factors such as the properties of \(a\) and whether \(n\) is even or odd.Hence, understanding the degree of a polynomial helps set expectations regarding the number of potential solutions but calls for a deeper investigation to ascertain the nature of these solutions.