Question

How many terms are there in each expression?$$x^{3}-2 x$$

Short answer

There are 2 terms in the expression \(x^{3}-2x\).

Step-by-step solution

Identify the Terms

Analyze the given expression and identify each term separated by a subtraction or addition operator. In the expression \(x^{3}-2x\), there are two separate terms, \(x^{3}\) and \(-2x\).

Count the Terms

Count the number of individual terms identified in Step 1. In the expression, there are two distinct terms, so the answer to how many terms are in the expression is two.

Key concepts

Identifying Terms in Algebra

Understanding how to identify terms in an algebraic expression is a foundational skill in algebra. Terms are the building blocks of algebraic expressions and are separated by addition (+) or subtraction (-) signs. For instance, in the expression x^{3}-2x, we note two distinct parts: x^{3} and -2x, which are the terms.

Each term can consist of numbers, variables, or a combination of both, possibly raised to a power. In our example, the first term x^{3} includes the variable x raised to the third power, and the second term -2x includes a coefficient (-2) multiplied by the variable x. The negative sign in front of the 2 indicates that this term is subtracted from the first.

When identifying terms, look for the plus or minus signs that serve as boundaries between them. But do remember, the coefficient of a term can be negative, which is an integral part of the term itself, as seen with -2x. It's crucial not to separate the negative sign from its term during identification.

Algebraic Expressions

Algebraic expressions are combinations of terms that can include numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation. Variables serve as placeholders for values and can represent unknowns or quantities that can change.

Expressions do not contain equality signs; that's what distinguishes them from equations. An example of an algebraic expression is x^{3}-2x, which we previously identified as containing two terms. Expressions can be simplified or manipulated through various algebraic operations to solve algebraic problems.

Understanding and manipulating algebraic expressions is essential as they are used to model real-world situations and solve problems. For example, if x represents the number of apples in a basket, the expression could be used to calculate the total weight of the apples if each apple's cube and weight are considered.

Subtraction of Polynomials

Subtraction is one of the basic operations you can perform on polynomials, which are algebraic expressions that include terms with variables raised to whole-number exponents. The process of subtracting polynomials is similar to combining like terms and involves lining up like terms—those with the same variables raised to the same powers—and then subtracting the coefficients.

Let's look at a basic example using the expression x^{3}-2x. If you were to subtract 3x^{2}+x from this expression, you would rearrange the terms to line up the like terms and then subtract: (x^{3}-2x)-(3x^{2}+x). Notice that in the second polynomial 3x^{2}+x, there isn't a term that matches x^{3}, so the x^{3} term remains unchanged.

One might argue that subtraction of polynomials is just addition with additional steps, as subtracting a polynomial is equivalent to adding the opposite of each of its terms. This approach can simplify the process and help avoid sign mistakes when combining terms.