Problem 30 - Elementary Algebra

Question

For the expressions in the following problems, write the number of terms that appear and then list the terms. $$ 17 a b^{2} $$

Step-by-Step Solution

Answer

Answer: There is 1 term in the expression \(17ab^2\), which is the whole expression itself, \(17ab^2\).

Step 1: Identify the terms

To identify the terms of an expression, look for each part separated by addition or subtraction symbols. In our expression, \(17ab^2\), there is only one term because it is not being added or subtracted by any other term. The term is \(17ab^2\).

Step 2: Count and list the terms

Since there is only one term in the expression, the number of terms is 1. The complete list of terms in the expression is: 1. \(17ab^2\) In conclusion, the given expression has only one term which is \(17ab^2\).

Key Concepts

Polynomial Terms

Understanding polynomial terms is essential when studying algebra. A polynomial is a mathematical expression consisting of variables (also called indeterminates), coefficients, and exponents that are combined using only addition, subtraction, multiplication, and non-negative integer exponents. The individual parts of a polynomial, separated by plus or minus signs, are known as terms. Each term is a product of a coefficient (a constant number) and the variables it contains raised to their respective powers.

For example, in the expression \( 17ab^2 \), there is a single term consisting of a coefficient (17) and variables (a and b) with an exponent (2) only on variable b. This term can be visually represented by a monomial because it has only one term. In polynomials with more than one term, like \( 3x^2 + 2x + 7 \), there are three terms, where \( 3x^2 \), \( 2x \), and 7 are each considered separate terms.

Identifying Terms in an Expression

To master algebra, one must be adept at identifying terms in an algebraic expression. Each term in an expression represents a separate component which could be combined with other terms through addition or subtraction. These terms are the building blocks of algebraic expressions and help us understand the structure of an equation or formula.

For instance, in our example, the expression \( 17ab^2 \) has only one term, which means there are no plus or minus signs present to indicate separation between terms. To identify terms, always look for these signs as indicators for where one term ends and another begins. If no such signs are visible, as with a standalone expression like \( 17ab^2 \), we determine that there is only one term within the expression. It is crucial not to overlook any terms, as this would result in inaccurate calculations or simplifications in algebraic tasks.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a specific value or a range of values. Algebraic expressions are fundamental to algebra and help in forming equations and inequalities.

Expressions can be as simple as a single term, like \( 17ab^2 \). They can also be more complex with multiple terms such as \( 4x - 5y + 3 \). We interpret these expressions by combining like terms and applying arithmetic operations. Remember, like terms are those that have the same variables raised to the same power, even if they have different coefficients. Breaking down algebraic expressions into their individual terms allows us to simplify or manipulate an equation with precision, ultimately solving for unknown variables or understanding the relationship between various mathematical elements.