Problem 116 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the terms. Then identify the coefficients of the variable terms of the expression. $$7 x^{4}+\sqrt{2} x^{2}-x$$

Step-by-Step Solution

Answer

The three terms in the given expression are \(7 x^{4}\), \(\sqrt{2} x^{2}\), and \(-x\), with their coefficients being \(7\), \(\sqrt{2}\), and \(-1\) respectively.

Step 1: Identifying the Terms

An expression is made up of terms, which are separated by plus (+) or minus (-) signs in an expression, so the given expression \(7 x^{4}+\sqrt{2} x^{2}-x\) has three terms: \(7 x^{4}\), \(\sqrt{2} x^{2}\), and \(-x\).

Step 2: Identifying Coefficients of Variable Terms

The numerical factor of a term with a variable is called the coefficient. For every term, now we will identify its coefficient. For the term \(7 x^{4}\), the coefficient of the variable \(x^{4}\) is \(7\). For the term \(\sqrt{2} x^{2}\), the coefficient of the variable \(x^{2}\) is \(\sqrt{2}\). For the term \(-x\), the coefficient of the variable \(x\) is \(-1\). The -1 is often not written, however a term \(-x\) is equivalent to \(-1*x\).

Key Concepts

Identifying Terms of an Expression

When dealing with algebraic expressions, one of the foundational skills is to identify its terms. Terms in an algebraic expression are the distinct elements that are separated by either addition (+) or subtraction (-) symbols. For example, in the expression \(7 x^{4}+\sqrt{2} x^{2}-x\), we have three separate terms: the first term is \(7 x^{4}\), the second term is \ \(\sqrt{2} x^{2})\ \), and the third term is \(\-x\).

Understanding each of these terms is crucial because it allows you to analyze and simplify expressions efficiently. Terms can consist of constants, coefficients, and variables, which we’ll explore further in the following sections.

Coefficients in Algebra

In algebra, coefficients play a key role in understanding the significance of variable terms in expressions. A coefficient is a numerical factor that multiplies a variable within a term. Taking our example expression \(7 x^{4}+\sqrt{2} x^{2}-x\), the coefficients are quite easy to identify once you understand the definition. The coefficient for the term \(7 x^{4}\) is \(7\), for \(\sqrt{2} x^{2}\) it's \(\sqrt{2}\), and for \(\-x\), the coefficient is -1, though it isn’t explicitly written.

It's important to remember that every variable term has a coefficient, even if it's not visible. For instance, the term \(x\) has an implied coefficient of \(1\), and \(\-x\) is really \(\-1\cdot x\). Recognizing coefficients correctly is vital for further algebraic operations such as multiplication or factoring.

Variable Terms in Polynomials

Variable terms in polynomials are the parts of the expression that contain variables raised to a power, such as \(x\), \(x^{2}\), \(x^{4}\), etc. In the expression \(7 x^{4}+\sqrt{2} x^{2}-x\), each term is a variable term with differing degrees (powers).

The individual degrees in our terms are 4, 2, and 1, corresponding to the exponents on the variable \(x\). Typically, variable terms indicate how many times a variable is used as a factor in the term. For example, \(x^{4}\) means \(x\) is multiplied by itself four times. When working with polynomials, it is crucial to identify the degree of each term to understand the expression's behavior, especially for tasks like graphing or solving equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a specific value or set of values. In our example, \(7 x^{4}+\sqrt{2} x^{2}-x\) is an algebraic expression with three terms, variables, and coefficients.

Expressions can be simplified, factored, or expanded, providing the groundwork for equations and more advanced algebraic problems. Algebraic expressions do not include equality or inequality signs; those are reserved for equations and inequalities. Practicing with expressions like these builds intuition for understanding algebraic principles and solving more complex problems in mathematics.