Problem 5 - Algebra for College Students

Question

Determine the degree of the given polynomials. $$5 x^{2}-7 x-2$$

Step-by-Step Solution

Answer

The degree of the polynomial is 2.

Step 1: Identify Each Term of the Polynomial

The given polynomial is \(5x^2 - 7x - 2\). It has three terms: \(5x^2\), \(-7x\), and \(-2\).

Step 2: Determine the Degree of Each Term

The degree of a term in a polynomial is determined by the exponent of the variable \(x\). For \(5x^2\), the degree is 2. For \(-7x\), the degree is 1. For \(-2\), which can be seen as \(-2x^0\), the degree is 0.

Step 3: Identify the Highest Degree

The degree of a polynomial is the highest degree from among its terms. Comparing the degrees found: 2, 1, and 0, the highest is 2.

Step 4: State the Degree of the Polynomial

Since the highest degree of any term in the polynomial is 2, the degree of the polynomial \(5x^2 - 7x - 2\) is 2.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental concept that determines many of its properties and behaviors. Simply put, the degree is the highest power of the variable present in the polynomial. In the polynomial \(5x^2 - 7x - 2\), the degree is 2 because the term \(5x^2\) contains the highest exponent, which is 2.

Understanding the degree of a polynomial helps in analyzing its graph, determining its end behavior, and predicting how it will interact with other functions. For example, a polynomial with a degree of 2, such as a quadratic, will graph as a parabola. Knowing the degree can also help in solving equations and inequalities involving polynomials. When comparing different polynomials, the degree provides a quick way to assess which will grow faster as the variable becomes very large.

Terms of a Polynomial

A polynomial is made up of terms, which are individual components separated by plus or minus signs. Each term is composed of a coefficient, a variable (such as \(x\)), and an exponent. In the polynomial \(5x^2 - 7x - 2\):

• The first term is \(5x^2\)
• The second term is \(-7x\)
• The third term is \(-2\)

Each term plays a role in the overall structure and value of the polynomial. The power of the variable in each term, indicated by the exponent, dictates the degree of that particular term. The sign and coefficient of the term influence its contribution when evaluating the polynomial for different values of \(x\). Breaking down polynomials into terms is a good practice for simplifying operations such as addition, subtraction, or factoring.

Exponents in Polynomials

Exponents in polynomials are the numbers that appear as superscripts in the terms and indicate the power to which the variable is raised. They tell us how many times to multiply the variable by itself. In our polynomial example, \(5x^2 - 7x - 2\), the exponents are critical for understanding its shape and behavior.

• In \(5x^2\), the exponent is 2, meaning \(x\) is squared.
• In \(-7x\), the exponent is 1, as any number raised to the power of 1 is itself.
• In \(-2\), the exponent is implicitly 0 because \(x^0 = 1\).

Exponents not only determine the degree of each term but also help us to identify the degree of the entire polynomial. They help in simplifying polynomials and are crucial for performing polynomial operations like differentiation and integration in calculus.