Question

Identify the degree of each polynomial. $$8 y^{4}+2 y-5 y^{6}+y^{3}+4$$

Short answer

Answer: The degree of the given polynomial is 6.

Step-by-step solution

Identify the degrees of individual terms

To do this, look at each term in the polynomial and find the power of y in that term. The power of y represents the degree of that specific term. 1. \(8y^4\): Degree 4 2. \(2y\): Degree 1 3. \(-5y^6\): Degree 6 4. \(y^3\): Degree 3 5. \(4\): Degree 0 (constant terms have a degree of 0)

Find the highest degree in the polynomial

Now that we have identified the degrees of each term, we need to find the highest degree present in the polynomial. In this case, the highest degree is 6 (associated with the term \(-5y^6\)).

State the degree of the polynomial

The degree of the given polynomial is 6, as the term with the highest power of y is \(-5y^6\).

Key concepts

Understanding Polynomials

Polynomials are one of the most fundamental concepts in algebra and are used extensively in mathematics and science. A polynomial is a mathematical expression consisting of variables (also called indeterminates), coefficients, and the operation of addition, subtraction, and multiplication. A polynomial can have any number of terms, each term being a product of a coefficient (a constant) and a variable raised to a non-negative integer power. For example, the expression 8y^4 + 2y - 5y^6 + y^3 + 4 has five terms and is a polynomial in variable y.

Polynomials appear in various forms and complexities, ranging from single-variable polynomials like x^2 + 3x + 4 to multivariable polynomials such as 3xy^2 - 7x^2y + 2y + 5. They are used to formulate mathematical models, approximate functions, and solve equations. Familiarizing yourself with the structure and composition of polynomials is key to understanding more advanced topics in algebra and calculus.

Determining the Degree of a Polynomial

The degree of a polynomial is a critical attribute that indicates the highest power of the variable within the polynomial. Specifically, it is determined by looking at the exponents of the terms and identifying the term with the highest exponent. The degree gives insight into the behavior of the polynomial function, especially concerning its growth and the number of zeros it may have.

For example, in the polynomial 8y^4 + 2y - 5y^6 + y^3 + 4, we begin by evaluating the degree of each individual term:

• 8y^4: Degree 4
• 2y: Degree 1
• -5y^6: Degree 6
• y^3: Degree 3
• 4: Degree 0 (since this is a constant term)

The term with the highest exponent is -5y^6, which means that the degree of the entire polynomial is 6. This highest degree influences the shape and number of turns in the graph of the polynomial function. Polynomials of higher degrees tend to have more complexity in their graphs.

Identifying Terms of a Polynomial

A term in a polynomial is an expression that is either a constant or a product of a constant (the coefficient) and one or more variables raised to non-negative integer powers. When identifying the terms of a polynomial, it is helpful to recognize that each term is separated by either a plus (+) or minus (-) sign. Each term in a polynomial represents a separate component, and together, all terms form the complete algebraic expression.

In the polynomial 8y^4 + 2y - 5y^6 + y^3 + 4, there are five terms, which are:

• 8y^4 (a monomial because it has a single term)
• 2y (also a monomial)
• -5y^6 (again, a monomial)
• y^3 (another monomial)
• 4 (a constant term, considered as a monomial with zero degree)

The coefficients in these terms are 8, 2, -5, 1 (implied), and 4, respectively. These terms can be manipulated individually, which is especially useful in operations like adding or subtracting polynomials where like terms are combined.