Problem 3 - Elementary Algebra

Question

List the terms in the following expressions. $$ 2 x y+6 x^{2}+(x-y)^{4} $$

Step-by-Step Solution

Answer

Question: Identify and list the terms present in the following expression: \(2xy + 6x^2 + (x-y)^4\) Answer: The terms present in the given expression are: 1. \(2xy\) 2. \(6x^2\) 3. \((x-y)^4\)

Step 1: Identify the terms

In the given expression, we can see three terms separated by "+" or "-". $$ 2xy + 6x^2 + (x-y)^4 $$ The terms are - \(2xy\), \(6x^2\), and \((x-y)^4\).

Step 2: List the terms

List out the identified terms as follows: 1. \(2xy\) 2. \(6x^2\) 3. \((x-y)^4\) These are the terms present in the given expression.

Key Concepts

Terms of an Expression

Understanding the 'terms of an expression' is a fundamental concept in algebra. A term is a single mathematical expression which may consist of constants, variables, or a mix of both, possibly including coefficients and powers. The expression 2xy + 6x^2 + (x-y)^4 contains three distinct terms separated by plus signs.

Each term in an algebraic expression represents a distinct component that can stand alone. For example, in the expression above, the coefficient 2 in the first term 2xy is multiplied by two variables, x and y. This differs from the second term, 6x^2, where the coefficient 6 is multiplied by x squared. The third term, (x-y)^4, is a binomial raised to the fourth power, which indicates repeated multiplication of the binomial by itself four times.

When listing the terms, ensure to include the whole term as it appears in the expression, including its coefficients and any exponentiated variables or expressions, as was correctly identified in the step-by-step solution.

Polynomials

Polynomials are algebraic expressions that consist of one or more terms. Each term in a polynomial is typically composed of variables and coefficients, with the variable parts raised to whole-number exponents, and the terms are combined using addition or subtraction. In our example, we are working with a polynomial: 2xy + 6x^2 + (x-y)^4.

A critical feature of polynomials is that each term only contains positive integer exponents: the expression 6x^2 confirms that with the exponent 2 on x. Furthermore, polynomials are often organized in descending order based on the degree of each term, which is the sum of the exponents on the variables. However, in the given expression, the terms have not been arranged by degree. If looking for a particular structure, we might rearrange them after expanding and combining similar terms. As polynomials can model a vast array of problems, recognizing and working with them effectively is key for progressing in mathematics.

Exponents

Exponents play a vital role in algebra. They tell us how many times to multiply a number by itself. For instance, in the term (x-y)^4, the exponent is 4, which signifies that the binomial x-y should be multiplied by itself four times. It's crucial to distinguish between the exponent of a single variable like x^2, where only x is squared, and the exponent of a binomial like (x-y)^4, affecting the whole expression within the parentheses.

Exponents can drastically change the value and complexity of an algebraic expression. If we were to expand (x-y)^4, the result would be a polynomial with multiple terms, each representing the product of applying the distributive property repeatedly. This expansion is beyond just squaring a binomial; it involves a pattern known as the binomial theorem or Pascal's triangle for higher powers. Exponents are not only foundational in algebra but also in various applications across mathematics and science. Understanding how to manipulate and evaluate them, therefore, is essential for any student's mathematical toolkit.