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Chapter 4: Problem 7

Determine whether the terms are like or unlike. $$4 x^{3}, 5 x^{3}$$

Short Answer

Expert verified
The terms \(4 x^{3}\) and \(5 x^{3}\) are like terms, as they have the same variable 'x' and the same exponent '3'.

Step by step solution

01

Identify Variables and Exponents

Look at the variables in both terms. In our case, the variable in both terms is 'x'. Similarly, check for the exponent. In this case, the exponent is '3' for both terms.
02

Compare Variables and Exponents

After identifying variables and exponents, the next step is to compare them. As in our terms both variables and exponents are the same, they are classified as like terms.

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Key Concepts

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

Understanding Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols, often in the form of letters, represent numbers and are used to formulate equations and expressions.
A key part of algebra is learning how to work with these symbols to solve problems.
  • It involves operations such as addition, subtraction, multiplication, and division.
  • Algebra serves as the foundation for more advanced topics in mathematics, such as calculus and linear algebra.

When you encounter terms in an algebraic expression, recognizing whether they are "like" or "unlike" is essential. This comparison allows for combination and simplification of expressions, which is a fundamental skill in algebra.
Role of Variables
Variables are symbols used in algebra to represent unknown values or numbers. The most common symbols used as variables are letters such as x, y, and z.
Each variable uniquely holds a value, and understanding how to manipulate these is crucial for solving equations.
Variables perform several important functions:
  • They allow generalization of mathematical relationships, which apply to many situations.
  • Variables can change or vary, making them flexible tools for formulation of equations.

In the exercise given, the variable is 'x', and correctly identifying it is critical to determine whether terms are like or unlike.
Understanding Exponents
Exponents are a way of expressing repeated multiplication of a number by itself. In algebra, exponents are used to describe the degree of a variable in a term. The exponent tells you how many times the base, usually found as a variable, is used in multiplication.
For example, in the term \( x^3 \), the base is \( x \) and the exponent is 3, meaning \( x \times x \times x \).
  • Exponents follow several key rules, such as the product rule: \( a^m \times a^n = a^{m+n} \).
  • Understanding exponents helps in simplifying expressions and solving equations.

In the original exercise, both terms have the same exponent, which is 3. This similarity helped to confirm they are "like" terms.
Introduction to Polynomials
Polynomials are mathematical expressions that consist of variables, coefficients, and exponents, combined through operations of addition, subtraction, and multiplication. A polynomial can have one or multiple terms.
Each term in a polynomial is known as a monomial, and these terms are classified as like or unlike.
  • The terms are like if they have the same variable with the same exponent.
  • Unlike terms cannot be combined because their variables or exponents are different.

Understanding polynomials is essential for solving, simplifying, and factoring in algebra. In the exercise, identifying like terms allowed for simplification and combination, crucial skills in learning to work with polynomials.

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Most popular questions from this chapter

If \(f(x)=x^{2}-2 x+3,\) find each value. $$f(5)$$

Describe how to determine the degree of a polynomial.

Simplify or solve as appropriate. $$(3 x+4)(2 x-2)-(2 x+1)(x+3)$$

Perform each division. Assume no division by \(0 .\) $$\frac{25 x^{2}-16}{5 x-4}$$

Classify each polynomial as a monomial, a binomial, a trinomial, or none of these. $$\frac{1}{2} x^{3}+3$$
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