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Chapter 2: Problem 353

In the following exercises, identify the terms in each expression. $$ 12 n^{2}+3 n+1 $$

Short Answer

Expert verified
12n^2, 3n, 1

Step by step solution

01

Identify the Polynomial Expression

Look at the given expression: 12n^2 + 3n + 1
02

Recognize the Components

Identify the different parts of the expression. The expression is composed of different terms separated by the addition sign (+).
03

List Each Term Separately

Break down the expression to list each term separately: 1. 12n^2 2. 3n 3. 1

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Expression
In algebra, a polynomial is a type of mathematical expression. It consists of variables, coefficients and exponents arranged in terms. The general form of a polynomial in one variable, say x, is given by: \text{a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0}\, where \(a_i\) are constants (coefficients) and \(n\) is a non-negative integer representing the exponent. In this form, \(a_nx^n\) is the leading term and \(a_0\) is the constant term. Polynomial expressions can vary in complexity, from simple to very complex. For instance, the expression given in the task, 12n^2 + 3n + 1, is a polynomial expression in the variable n.
Terms in Algebra
Terms in algebra refer to the individual components of an expression separated by addition or subtraction signs. Each term consists of a numerical coefficient and one or more variables raised to an exponent. Consider this example of a polynomial expression: 12n^2 + 3n + 1
Here鈥檚 how we identify and list each term:
  • 12n^2: This is a term where 12 is the coefficient, and n is raised to the power of 2
  • 3n: In this term, 3 is the coefficient, and n is raised to the power of 1
  • 1: This is a constant term with no variable attached
Each of these separated parts by the 鈥�+鈥� sign are individual terms in this expression. Recognizing terms correctly is vital for solving and simplifying polynomial equations.
Breaking Down Expressions
Breaking down polynomial expressions into individual terms helps in understanding and solving algebraic problems. Let's apply the step-by-step approach to the given exercise: 12n^2 + 3n + 1
  • Step 1: Identify the expression.
  • Step 2: Recognize the components. Each term is separated by a 鈥�+鈥� sign.
  • Step 3: List each term separately:
  • First term: 12n^2 (12 is the coefficient and n is raised to the power of 2)
  • Second term: 3n (3 is the coefficient and n is raised to the power of 1)
  • Third term: 1 (This is a constant term)
By breaking down expressions into smaller parts, handling and solving problems becomes more manageable.

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