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Chapter 0: Problem 4

Identify the terms of the algebraic expression.\(4 x^{3}+x-5\)

Short Answer

Expert verified
The terms of the algebraic expression \(4 x^{3}+x-5\) are \(4 x^{3}\), \(x\), and \(-5\).

Step by step solution

01

Identify The Terms

Looking at the expression, we can see that it is a sum of three terms: \(4 x^{3}\), \(x\), and \(-5\). Notice that '-5' is considered as one term, not two. We include the negative sign in the term.

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Key Concepts

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

Polynomial Terms
In mathematics, specifically in algebra, an expression like \(4x^{3} + x - 5\) is called a polynomial. A polynomial is a series of terms, which are added or subtracted together. Each part that is separated by a plus \((+)\) or minus \((-\)) sign is considered a term. Let's break this concept down using our example polynomial.

In the expression \(4x^{3} + x - 5\), we can identify three distinct polynomial terms as follows:
  • \(4x^{3}\): This is the first term which combines a number and a variable raised to a power.
  • \(x\): This is the second term, a variable with no visible number in front, often implying multiplication by 1.
  • \(-5\): A constant term which stands on its own without a variable, but includes the negative sign to maintain its value.
Understanding terms helps in simplifying and solving polynomials and is a key skill in algebra.
Coefficients
In algebraic expressions, coefficients are the numerical factors in a term that contain variables. They play an essential role in defining how much a variable contributes to the term.

Let's look at the polynomial \(4x^{3} + x - 5\) and identify the coefficients of each term.
  • In the term \(4x^{3}\), the number 4 is the coefficient. It indicates that the variable \(x^{3}\) is multiplied by 4.
  • For the term \(x\), the coefficient is implicitly 1, as it is often not written but understood that \(x = 1x\).
  • However, for the term \(-5\), there is no variable, so it is not considered to have a coefficient in the traditional sense. Instead, it is a constant term.
Recognizing coefficients allows for the comparison and manipulation of polynomial expressions, crucial for solving algebraic equations.
Variables
Variables are symbols used in mathematics to represent unknown or changeable values. In algebra, they are usually denoted by letters such as \(x\), \(y\), or \(z\).

In the expression \(4x^{3} + x - 5\), variables act as placeholders for numbers and allow the algebraic manipulation of expressions. Here's how they appear in our polynomial:
  • The term \(4x^{3}\) contains the variable \(x\), raised to the third power. This notation indicates that the variable affects the term's value significantly depending on its exponent.
  • In the term \(x\), the variable appears without an exponent, which default to an exponent of 1, meaning \(x^1 = x\).
  • The term \(-5\) does not include any variable and stands as a constant, unaffected by the variables in the other terms.
Understanding variables is vital as they allow the representation and solution of general problems in algebra, forming the foundation on which equations are solved.

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