Question

State the degree of each term.$$5 x^{2} y^{3}$$

Short answer

The degree of the term \(5x^2y^3\) is 5.

Step-by-step solution

Understand Degree of a Term

The degree of a term in a polynomial is the sum of the exponents of the variables in the term.

Identify the Exponents

In the term \(5x^2y^3\), the exponents of the variables are 2 for \(x\) and 3 for \(y\).

Calculate the Degree

Add the exponents of the variables to find the degree of the term: \(2 + 3 = 5\).

Key concepts

Polynomial

Polynomials are mathematical expressions comprising variables, coefficients, and exponents, arranged in terms of decreasing powers. Each part of a polynomial is known as a term, and these terms are separated by addition or subtraction operators. Polynomials can have a single term, like \(3x^4\), known as a monomial, or multiple terms, such as \(4x^3 - 2x^2 + x - 5\), which is called a quartic polynomial due to its highest degree of 4.

Understanding polynomials is fundamental in algebra and appears in various applications, from basic math to advanced calculus. The degree of a polynomial is determined by the term with the highest degree, which is the sum of the exponents in that term and indicates the polynomial's 'size' in terms of its power.

• If a polynomial has one variable, its degree is the highest exponent of that variable.
• For multiple variables, add up the exponents of each variable within a term to determine that term's degree, and the highest resulting degree among all terms is the degree of the polynomial.

Exponents

Exponents, also known as powers, are a shorthand notation to represent repeated multiplication of a number by itself. The exponent is a small number written above and to the right of a base number, indicating how many times the base is multiplied by itself. For example, in \(3^5\), the base is 3 and the exponent is 5, which means \(3 \times 3 \times 3 \times 3 \times 3\), or 243.

Exponents play a critical role in determining the degree of a term in a polynomial. When variables are involved, as in \(x^3\), the exponent indicates the degree of that variable. In the case of multiple variables multiplied together like \(x^2y^3\), each variable's exponent contributes to the degree of the term.

Variables

Variables are symbols used in mathematics to represent unknown values or values that can change. They are typically denoted by letters such as \(x\), \(y\), or \(z\). In the context of polynomials, variables are combined with coefficients and exponents to create terms.

In a polynomial, the variables can be raised to different exponents, affecting the overall degree of the polynomial. For example, in the term \(7x^4y^2\), the variables are \(x\) and \(y\). The degree of this term is 6, which is the sum of the exponents 4 and 2. The interplay between coefficients, variables, and exponents defines the structure and properties of a polynomial, including its degree. When determining the degree, only the exponents of variables are summed; coefficients do not influence the degree of a term or a polynomial.