Verified Answer :

  • Define a constant as a term in a polynomial that does not contain any variables.
  • Analyze the expression \(15x^2 + 2x + 9\) to find its components: \(15x^2\), \(2x\), and \(9\).
  • Identify that \(9\) is the only term that lacks a variable.
  • The constant is \(\boxed{9}\).

Explanation

  1. Analyzing the Expression and Defining Constants

To identify the constant in the expression \(15x^2 + 2x + 9\), we first define what a constant term represents in a polynomial. A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. A constant term is a term that does not contain any variables and therefore remains the same, regardless of the value of the variable.

  1. Breaking Down the Expression

The given expression is \(15x^2 + 2x + 9\). We can identify the individual terms:

  • The first term is \(15x^2\), where \(15\) is the coefficient and \(x\) is the variable.
  • The second term is \(2x\), where \(2\) is the coefficient and \(x\) is the variable.
  • The third term is \(9\), which is a numerical value without any variable.
  1. Identifying the Constant Term

By comparing the terms, we observe that the terms \(15x^2\) and \(2x\) depend on the value of the variable \(x\). The term \(9\), however, does not contain a variable and is, by definition, the constant term of the expression. Comparing this with the provided options (2, 24, 9, 15), it is clear that 9 is the correct choice.

Examples

In everyday life, constants are common in cost calculations. For example, if you pay a monthly internet bill, the cost might be modeled by a linear expression like \(C = 50 + 10x\), where \(C\) is the total cost, \(50\) is the fixed base fee (a constant), and \(10x\) represents an additional charge of \(\$10\) per extra gigabyte used. The base fee of \(\$50\) is independent of your data usage, similar to the constant term in our polynomial.

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