Questions and Answers
In the expression $x + 3x + 2x^2 - 5x^2 + 4x^3$, which terms can be combined as like terms?
$x$ and $3x$
If you have the expression $3x + 2x^2 + 7y - 5y^3$, which pairs of terms can be combined as like terms?
$3x$ and $2x^2$
When simplifying the expression $1.5x + 2x + 3x - 1.2x$, what is the correct result?
$6.3x$
Why is it important not to combine unlike terms in algebraic expressions?
Signup and view all the answers
What fundamental skill in algebraic expressions can help simplify problems and understand relationships between variables?
Signup and view all the answers
In the expression $2x + 3x - 5x^2 - 4x^2$, which terms are like terms?
Signup and view all the answers
What is the result of combining the like terms $5x - 3x + 4x$?
Signup and view all the answers
Which of the following pairs of terms in $3x^2 - 2x^2 + 5x - x$ are like terms?
Signup and view all the answers
Why are the terms $3y^2$ and $-5y^3$ not like terms?
Signup and view all the answers
If we have $2a + 3a - 4a + a$, what is the simplified expression?
Signup and view all the answers
Study Notes
Algebraic Expressions: Combining Like Terms
Algebraic expressions are fundamental building blocks in the world of algebra, where we manipulate symbols and numbers to solve problems and represent relationships. One of the essential techniques in working with algebraic expressions is combining like terms. In this section, we'll explore how to combine like terms, understand their significance, and appreciate their role in simplifying more complex expressions.
What are Like Terms?
Like terms are expressions with the same variable raised to the same power. For instance, in the expression (x + 3x + 2x^2 - 5x^2 + 4x^3), the terms (x) and (3x) are like terms because they both involve the variable (x) raised to the power of 1. Similarly, the terms (2x^2) and (-5x^2) are like terms because they both involve the variable (x) raised to the power of 2. However, the terms (x) and (2x^2) are not like terms because they involve different powers of (x).
Combining Like Terms
To combine like terms, we follow a simple rule: add the coefficients of the like terms while keeping the same variable and exponent. For example, in the expression (x + 3x + 2x^2 - 5x^2 + 4x^3), we can combine like terms by adding the coefficients of the (x) terms and the coefficients of the (x^2) terms:
[x + 3x + 2x^2 - 5x^2 + 4x^3 = (x + 3x) + 2x^2 - 5x^2 + 4x^3]
Simplifying this expression gives:
[x + 3x + 2x^2 - 5x^2 + 4x^3 = 4x + 2x^2 - 5x^2 + 4x^3 = 2x^2 + 4x^3]
Real-world Applications
Combining like terms can help simplify more complex expressions, making it easier for us to solve problems and understand relationships between variables. For example, consider a problem where we are given the expression (1.5x + 2x + 3x - 1.2x) representing the cost of three items in a shopping cart. By combining like terms, we can simplify the expression:
[1.5x + 2x + 3x - 1.2x = (1.5 + 2 + 3 - 1.2)x = 6.3x]
Now, we can see that the simplified expression represents the total cost of buying three items, each costing (6.3) times the original variable (x).
Challenges
While combining like terms can simplify many expressions, it's important to remember that we should not combine terms that are not like terms. For instance, in the expression (3x + 2x^2 + 7y - 5y^3), we cannot combine the terms involving (x) and the terms involving (y), because they are not like terms. Incorrectly combining unlike terms can create errors and affect the accuracy of our calculations.
In conclusion, combining like terms is a fundamental skill in working with algebraic expressions. By understanding this concept and its applications, we can simplify expressions, solve problems, and better understand relationships between variables.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of combining like terms in algebraic expressions with this quiz. Explore the significance of like terms, learn how to simplify expressions, and understand the importance of not combining unlike terms. Practice solving problems involving combining like terms to strengthen your algebra skills.
