Problem 4 - Elementary Algebra

Question

For Problems \(1-8\), determine the degree of each polynomial. $$ 8 x^{2} y^{2}-2 x y^{2}-x $$

Step-by-Step Solution

Answer

The degree of the polynomial is 4.

Step 1: Understanding the Expression

The polynomial given is \(8x^2y^2 - 2xy^2 - x\). In this polynomial, each term consists of products of constants and variables which have exponents. Our goal is to determine the degree of the polynomial.

Step 2: Determine the Degree of Each Term

Identify the degree of each term by adding the exponents of the variables in that term. For the term \(8x^2y^2\), the degree is \(2 + 2 = 4\). For \(-2xy^2\), the degree is \(1 + 2 = 3\). For \(-x\), the degree is \(1\).

Step 3: Identify the Degree of the Polynomial

The degree of a polynomial is the highest degree of its terms. From our calculations, the degrees of the terms are 4, 3, and 1. Hence, the degree of the polynomial \(8x^2y^2 - 2xy^2 - x\) is 4.

Key Concepts

Polynomial Terms

A polynomial is essentially a sum of terms, where each term consists of a coefficient—a constant—and one or more variables raised to whole-number exponents. Let us break down the notion of "terms" within a polynomial: each "term" is a building block of the expression, separated by a plus or minus sign. For instance, in the polynomial \(8x^2y^2 - 2xy^2 - x\), there are three terms: \(8x^2y^2\), \(-2xy^2\), and \(-x\). Each of these terms plays an integral role in defining the polynomial.- **Coefficient**: This is the numerical part before the variables. In \(8x^2y^2\), the number 8 is the coefficient.- **Variable**: Variables such as \(x\) and \(y\) are placeholders for numbers.- **Exponents**: These are powers that denote how many times a variable is multiplied by itself.To understand the degree of a polynomial makes it essential to know the degree of its terms, as the highest degree among the individual terms gives the overall polynomial degree.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operation symbols. They form the basis of algebra, allowing us to represent and work with mathematical relationships abstractly. Understanding algebraic expressions is key to grasping how polynomials operate and are manipulated.- **Combination of Elements**: An algebraic expression, such as \(8x^2y^2 - 2xy^2 - x\), combines constants, coefficients, variables, and operations (like addition and subtraction).- **Evaluating Expressions**: This involves substituting values for variables and performing operations to simplify or solve the expression.- **Simplifying**: It means reducing the expression to a more straightforward or more manageable form without changing its value, often by combining like terms or factoring.Algebraic expressions are foundational to learning about polynomials, as they help understand how terms are added and subtracted within a polynomial.

Exponents in Algebra

Exponents are crucial components of algebra and polynomials. They indicate how many times a base number or variable is multiplied by itself. Understanding exponents is essential to finding the degree of polynomials.- **Notation**: An exponent is written as a superscript, such as \(x^2\), which means \(x\) multiplied by itself.- **Rules of Exponents**: Basic rules include the product of powers (\(x^a \cdot x^b = x^{a+b}\)), and power of a power (\((x^a)^b = x^{a\cdot b}\)). These rules help in simplifying expressions and finding degrees of terms.- **Degree of a Term**: To find the degree of a term in a polynomial, sum the exponents of all variables in that term. For example, the degree of \(8x^2y^2\) is \(2 + 2 = 4\), since both \(x\) and \(y\) are raised to the power of 2.By mastering the concept of exponents, one can efficiently determine the degree of terms, which is a stepping stone to fully understanding the degree of a polynomial itself.