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Chapter 2: Problem 90

In the following exercises, list the terms in the given expression. $$11 x^{2}+8 x+5$$

Short Answer

Expert verified
The terms are \(11x^{2}\), \(8x\), and \(5\).

Step by step solution

01

Identify the expression

The given expression is \(11x^{2} + 8x + 5\). Recognize that it is a polynomial in the variable \(x\).
02

Break down the expression into terms

A term in an algebraic expression is a single mathematical expression. In \(11x^{2} + 8x + 5\), each part separated by a plus (\(+\)) sign is a term.
03

List the terms

The terms in the expression \(11x^{2} + 8x + 5\) are: \(11x^{2}\), \(8x\), and \(5\).

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Key Concepts

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

Algebraic Expressions
Understanding algebraic expressions is key in the realm of algebra. An algebraic expression is a combination of numbers, variables, and arithmetic operations (like addition or multiplication). For example, in the expression 11虫虏 + 8x + 5, you have numbers (11, 8, 5), the variable x, and the operations of addition and multiplication. To grasp this better, remember the following:
  • Numbers within the expression are called constants.
  • Variables are symbols that can represent different values.
  • Operations denote how these numbers and variables interact with each other.
Algebraic expressions help us to write complex mathematical ideas concisely and solve equations to find an unknown variable's value.
Polynomial
Let's explore a specific type of algebraic expression - the polynomial. A polynomial is an expression made up of multiple terms, where each term includes a variable raised to a non-negative integer exponent. Consider the polynomial provided in the exercise: 11虫虏 + 8x + 5. It's a polynomial in the variable x. Here is why:
  • Each term follows the form ax^n, where a is a constant coefficient and n is a non-negative integer.
  • This particular polynomial has three terms.
  • Polynomials can be classified by their degree (the highest exponent). In our polynomial, the highest degree is 2 (coming from 11虫虏).
Polynomials are foundational in algebra because many algebraic operations, functions, and equations are built upon them.
Terms in Polynomial
Understanding the terms in a polynomial is crucial for solving problems. A term is a single standalone mathematical expression within a polynomial. It can include a coefficient (a number), a variable, and an exponent. In the polynomial 11虫虏 + 8x + 5, let's break down each term:
  • 11虫虏: This term has a coefficient of 11 and a variable x raised to the power of 2.
  • 8x: Here, the coefficient is 8, and the variable x is to the power of 1 (though we don鈥檛 always write the 1).
  • 5: This is a constant term without any variable.
Remember, identifying and understanding each term allows us to perform operations like addition, subtraction, and multiplication on polynomials easily. Recognizing the structure of these terms will help simplify and solve complex algebraic equations.

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