Question

State the degree of each term.$$4 y^{2}$$

Short answer

The degree of the term \(4y^2\) is 2.

Step-by-step solution

Identify the Terms

First, identify the individual terms in the expression. In this case, there is only one term, which is \(4y^2\).

Determine the Degree of the Term

The degree of a term in a polynomial is the sum of the exponents of the variables in the term. For \(4y^2\), the exponent of \(y\) is 2, so the degree of the term is 2.

Key concepts

Understanding Algebra

Algebra is a branch of mathematics that often deals with symbols and the rules for manipulating those symbols; these symbols represent quantities without fixed values, known as variables. Think of algebra as a scale where equations are balanced by maintaining equality. The flexibility that variables provide allows us to solve problems that involve unknown quantities.

When working with algebra, it's crucial to understand terms like 'term,' 'coefficient,' 'variable,' and 'constant.' In the expression \(4y^2\), for instance, \(4\) is the coefficient, \(y\) is the variable, and the whole \(4y^2\) is considered a term. Algebra teaches us how to handle these components to simplify and solve mathematical problems, such as finding the degree of polynomials.

Exponents in Polynomials

Exponents play a crucial role in understanding polynomials. An exponent in a polynomial indicates how many times a number or variable is multiplied by itself. For example, in \(y^2\), the exponent is \(2\), meaning \(y\) is multiplied by itself once (\(y \cdot y\)).

In the context of polynomials, the degree of the polynomial is determined by the highest power (exponent) of the variable in any term. Therefore, if we take \(4y^2+z^3\) as a polynomial, despite there being two terms, the degree of the polynomial is \(3\) because the highest exponent is seen in the term \(z^3\). Deciphering the degree of a polynomial is important as it gives us information about the behavior of the graph of the polynomial and the maximum number of possible intersections with the x-axis, which are the graph's roots.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operations, and sometimes grouping symbols that stand for a particular quantity. Unlike equations, expressions do not contain an equals sign. Instead, they're used to represent values that can be simplified or evaluated given certain values for their variables.

For example, the expression \(4y^2\) represents a specific quantity that can change depending on the value of \(y\). When we talk about simplifying or evaluating expressions, we follow the order of operations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction (PEMDAS). By mastering these fundamental operations, one can effectively operate within the field of algebra and tackle various mathematical challenges.