Problem 9 - Elementary Algebra

Question

Classify the following equations in terms of their degree. $$ 5 x^{7}=3 x^{5}-2 x^{8}+11 x-9 $$

Step-by-Step Solution

Answer

Answer: The given equation is an 8th-degree equation.

Step 1: Identify the terms containing the variable x

First, look at the given equation and identify all the terms containing the variable x: $$ 5x^{7} = 3x^{5} - 2x^{8} + 11x - 9 $$ We have four terms containing x: \(5x^{7}\), \(3x^{5}\), \(-2x^8\), and \(11x\).

Step 2: Find the power of x in each term

Next, we need to find the power of x in each term. In the given equation, these are: $$ 5x^{7} → 7 \\ 3x^{5} → 5 \\ -2x^{8} → 8 \\ 11x → 1 $$

Step 3: Determine the highest power of x

Now, determine the highest power of x among the four terms identified. In this case, it's easy to see that the highest power of x is 8 in the term \(-2x^{8}\).

Step 4: Classify the equation in terms of degree

Since the highest power of x is 8, we can classify the equation as having a degree of 8, making it an 8th-degree equation.

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is fundamental to categorizing and solving polynomial equations. Simply put, the degree of a polynomial is determined by the highest power of the variable present in the equation. Consider a polynomial in one variable such as x. Each term in a polynomial takes the form of a_nxⁿ, where a_n are coefficients and n represents the powers of x.

In the given exercise, we're presented with the polynomial equation \(5 x^{7}=3 x^{5}-2 x^{8}+11 x-9\). To find its degree, we look for the term with the highest power of x, which in this case is \(x^{8}\). Hence, the degree of the polynomial is 8, making this polynomial an 8th-degree polynomial.

It's important to note that polynomials are named based on their degree. For instance, a 2nd-degree polynomial is called a quadratic, while a 3rd-degree polynomial is named cubic. Knowing the degree helps in predicting the behavior of the polynomial's graph and the number of possible roots or intersections with the x-axis.

Polynomial Terms

The terms of a polynomial are the individual components separated by addition or subtraction signs. Each term consists of a variable raised to a power and multiplied by a coefficient. In our example, the terms of the polynomial \(5x^{7} = 3x^{5} - 2x^{8} + 11x - 9\) are \(5x^{7}\), \(3x^{5}\), \(2x^{8}\), \(11x\), and \(9\). It's worth mentioning that the constant term, in this case \(9\), is also considered a polynomial term, and its variable is assumed to have a power of 0.

Recognizing and working with individual terms is essential for basic operations such as addition, subtraction, and especially when differentiating or integrating polynomials. It's also useful in understanding the polynomial's structure, which informs strategy when solving or factoring the polynomial. When classifying polynomials by the number of terms, we utilize particular nomenclature—'monomial' for single-term, 'binomial' for two terms, and 'trinomial' for three terms, and so forth.

Power of a Variable

In mathematics, the power of a variable indicates how many times that variable is used in a multiplication. It's expressed as an exponent. For instance, in the term \(x^{3}\), the number 3 is an exponent, and it means \(x\) is multiplied by itself three times: \(x\) * \(x\) * \(x\).

In our exercise, we analyze the powers of the variable x in each term. For \(5x^{7}\), the power is 7; for \(3x^{5}\), it's 5; the term \(11x\) actually has an implied power of 1 (since \(x\) is the same as \(x^{1}\)), and the constant term \(9\) is equivalent to \(9x^{0}\), as any number raised to the power of 0 is 1.

Considering powers is crucial, particularly when performing operations such as multiplying or dividing polynomials, as the powers of variables must be handled according to exponent rules. It affects both the simplification process and the outcome when solving polynomial equations.