Problem 2 - Algebra for College Students

Question

Determine the degree of the given polynomials. $$-5 x^{2} y^{2}-6 x y^{2}+x$$

Step-by-Step Solution

Answer

The polynomial's degree is 4.

Step 1: Understanding Monomials in the Polynomial

The given polynomial is \[-5x^2y^2 - 6xy^2 + x.\]This polynomial is composed of three terms: \(-5x^2y^2\), \(-6xy^2\), and \(+x\). Each term is called a monomial. We will find the degree of each monomial separately.

Step 2: Determine the Degree of Each Monomial

For each monomial, the degree is the sum of the exponents of all the variables in the term. - In \(-5x^2y^2\), the exponents are \(2\) for \(x\) and \(2\) for \(y\). Degree = \(2 + 2 = 4\).- In \(-6xy^2\), the exponents are \(1\) for \(x\) and \(2\) for \(y\). Degree = \(1 + 2 = 3\).- In \(+x\), the exponent is \(1\) for \(x\). Degree = \(1\).

Step 3: Determine the Degree of the Polynomial

The degree of the entire polynomial is defined as the largest degree from all the individual monomials. From Step 2, the degrees of the monomials are \(4\), \(3\), and \(1\).Therefore, the degree of the polynomial \(-5x^2y^2 - 6xy^2 + x\) is \(4\).

Key Concepts

Monomials

A monomial is a single term in an algebraic expression made up of numbers and variables that are multiplied together. In mathematics, each term you see in a polynomial can be considered a monomial. It can be something simple like \(3x\) or more complex, such as \(-5x^2y^2\).
A few important points about monomials:

• They can include numbers, variables, or products of numbers and variables.
• Coefficients are the numbers in front of the variables. In \(3x\), 3 is the coefficient.
• Each monomial is separated by addition or subtraction signs in an expression.

When working with polynomials, understanding each monomial's structure helps to determine the degree effectively.

Exponents

Exponents are the small numbers written at the top right corner of a variable. They indicate how many times the variable is multiplied by itself. For example, in the expression \(x^2\), the 2 is an exponent showing that \(x\) is multiplied by itself: \(x \times x\).
Key characteristics of exponents include:

• They help define the degree of a variable within a monomial.
• Multiple variables can have their own exponents within a single monomial, such as \(-5x^2y^2\).
• Adding the exponents of all variables gives you the monomial’s degree.

Understanding exponents is crucial, as it plays a big role in determining the degree of both monomials and polynomials.

Variables in Polynomials

Variables are symbols used to represent numbers and are an essential part of polynomials. Components such as \(x\) and \(y\) are often used. These variables can have different values depending on the context of the problem.
To understand polynomials:

• Note that each term (or monomial) in the polynomial consists of variables raised to certain powers.
• Each variable’s exponent contributes to the term's degree within the expression.
• Variables may appear in various combinations. For example, in \(-5x^2y^2 - 6xy^2 + x\), both \(x\) and \(y\) appear in different arrangements.

Recognizing how variables interact with their exponents is crucial for calculating the polynomial degree.

Algebraic Expressions

An algebraic expression is comprised of numbers, variables, and arithmetic operations (such as addition, multiplication, and subtraction). It can range from simple monomials to complex expressions with several terms.
Important aspects of algebraic expressions include:

• Components like \(-5x^2y^2\) are part of wider expressions and contribute to the overall calculation of properties like degree.
• They do not include an equation sign—that feature distinguishes them from algebraic equations.
• Expressions allow us to generalize numbers and variables, providing a platform for performing calculations across various values.

Understanding algebraic expressions is foundational for breaking down the complexities of polynomials and calculating characteristics such as degree.