Problem 113 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the terms. Then identify the coefficients of the variable terms of the expression. $$-2 x^{2}+11 x+3$$

Step-by-Step Solution

Answer

The terms of the expression \(-2 x^{2}+11 x+3\) are \(-2 x^{2}\), \(11 x\), and \(3\). The coefficients of the variable terms are \(-2\) for \(x^{2}\) and \(11\) for \(x\).

Step 1: Identify the terms

In an algebraic expression, the terms are the parts of the expression which are separated by '+' or '-' symbols. In the given expression \(-2 x^{2}+11 x+3\), the terms are \(-2 x^{2}\), \(11 x\), and \(3\).

Step 2: Identify coefficients of variable terms

The coefficients are the numeric factors of variable terms. Here, in the term \(-2 x^{2}\), the coefficient of \(x^{2}\) is \(-2\), and in the term \(11x\), the coefficient of \(x\) is \(11\). The term \(3\) is a constant and does not have a coefficient.

Key Concepts

Identifying Terms in Algebra

Understanding algebraic expressions begins with being able to identify individual terms. In the world of algebra, a term is considered a single mathematical element that can be a number, a variable, or the product of numbers and variables. These terms are the building blocks of algebraic expressions, and they are typically separated by addition '+' or subtraction '-' signs.

For example, let's examine the expression \( -2 x^{2} + 11 x + 3 \). This expression contains three separate terms: \( -2 x^{2} \), \( 11 x \), and \( 3 \). By identifying each term, one lays the groundwork for further operations, such as simplification or evaluation of the expression.

Coefficients of Variables

Once terms are identified in an algebraic expression, the next step is to understand coefficients. Coefficients are the numbers that multiply the variable part of the terms, providing a measure of how much the variable is 'scaled' by in the term.

In the expression \( -2 x^{2} + 11 x + 3 \), the coefficient of the first term \( -2 x^{2} \) is \( -2 \) and the coefficient of the second term \( 11 x \) is \( 11 \). The coefficients can be positive, negative, or even fractional numbers that precede the variables, indicating the strength of the contribution of each variable term within an expression.

Polynomial Terms

Algebraic expressions that involve terms with variables raised to whole number exponents are known as polynomials. Each term of a polynomial has a variable (like \( x \) or \( y \) or another letter) raised to a non-negative integer exponent.

The expression \( -2 x^{2} + 11 x + 3 \) exemplifies a polynomial that has three terms. The term \( -2 x^{2} \) contains the variable \( x \) raised to the second power, which classifies this term as a quadratic. The term \( 11 x \) is a linear term because \( x \) is raised to the power of one. Understanding each term's degree - which is the exponent on the variable - is crucial in identifying the overall structure and behavior of the polynomial.

Constant Term in Algebra

A constant term in an algebraic expression is a term that does not involve any variables; essentially, it is a number that stands alone. In algebraic expressions, constant terms can be identified as those without a variable and they provide a baseline value for the expression.

Looking back to our example, \( -2 x^{2} + 11 x + 3 \), the number \( 3 \) is the constant term. It represents the part of the expression that will not change regardless of the value of \( x \). The constant term often plays a significant role in various algebraic procedures such as factoring or solving equations because it impacts the value of the entire expression when the variable terms equal zero.