Problem 104 - Introductory and Intermediate Algebra for College Students 4th

Question

Explain how to simplify \(4 x^{2}+6 x^{2} .\) Why is the sum not equal to \(10 x^{4} ?\)

Step-by-Step Solution

Answer

The simplified expression is \(10x^2\) and not \(10x^4\) because in the addition of the two like terms \(4x^2\) and \(6x^2\), we keep the exponent the same and simply add the coefficients. The concept of powers applies to multiplication and division but not for addition and subtraction.

Step 1: Identify like terms

First identify like terms in the given expression. When two terms have the same variable and exponent, they are defined as like terms. In this case, \(4x^2\) and \(6x^2\) are like terms.

Step 2: Simplify like terms

To simplify like terms, sum the coefficients while keeping the variable and exponent the same. In this case, the coefficient of \(x^2\) in \(4x^2 + 6x^2\) is \(4+6 = 10\). So, \(4x^2 + 6x^2\) simplifies to \(10x^2\).

Step 3: Understand why the sum is not \(10x^4\)

The sum is \(10x^2\) and not \(10x^4\) because in the addition of like terms, one does not multiply the exponents. When we add \(4x^2\) and \(6x^2\), we are adding two quantities that both include \(x\)'s exponent as \(2\). Multiplying the exponents would turn the equation into a product of powers, which is not the case here.

Key Concepts

Like Terms in Algebra

When working with algebraic expressions, understanding the concept of 'like terms' is crucial for simplification. These are terms that have the exact same variables raised to the same power. Imagine you have a few apples and someone gives you more apples; you simply end up with a larger number of apples. The same principle applies to like terms.

Let's think of the variables as containers and the exponents as labels on those containers. If the labels match perfectly, we can put the contents together. For instance, in the problem \(4x^2 + 6x^2\), both terms have the container labeled \(x^2\). Since they are identical, we can combine them by adding the numerical coefficients, which results in \(10x^2\).

Why do we just add the numbers in front and not change the exponent? Because the base and the exponent represent the type and size of each container. If we change those, we're no longer dealing with the same type of contents. As a result, simplification of like terms in algebra entails combining these matched sets without altering the 'labels' or exponents.

Addition of Polynomials

Adding polynomials is like organizing a large collection of various items into groups. The process involves identifying like terms, as discussed previously, and then adding them together. Consider polynomials as sets of multiple terms where each term can be thought of as a unique variety of items.

In the example \(4x^2+6x^2\), our polynomial is made up of two terms that are like terms. When adding polynomials, we work systematically, combining like terms to simplify the expression as much as possible. We look for terms with the same variables and exponents and merge them together by adding their coefficients.

It’s important to note that if we're doing the addition properly, the variables and their exponents remain unchanged; only the coefficients are summed up. This follows the fundamental principle of addition in algebra, which preserves the structure of the polynomial while combining quantities.

Exponents and Powers

Understanding exponents and powers is essential for correctly simplifying algebraic expressions. An exponent tells us how many times to multiply a number by itself. For example, \(x^2\) indicates that \(x\) should be multiplied by itself once, resulting in \(x*x\).

It is important to realize that operations with exponents follow specific rules. When we add terms with exponents, like \(4x^2 + 6x^2\), we are not performing an operation that requires changing the exponent. The sum \(10x^2\) maintains the exponent '2' because the operations of addition and multiplication (which exponents essentially represent) are distinct. Had we multiplied two terms with exponents, for example \(x^2 * x^2\), we would then apply the 'product of powers' rule, adding the exponents to get \(x^4\).

The confusion might arise from mixing up the rules for the addition of polynomials and the multiplication of powers. Remember, when adding or subtracting, the exponents remain untouched as long as we are combining like terms.