Simplify The Expression:$\[ 28x^2 - 15x^5yz^2 \\]

Feb 28, 2025 by ADMIN 50 views

Simplifying Algebraic Expressions: A Step-by-Step Guide

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the given expression: $28x^2 - 15x^5yz^2$ . We will break down the process into manageable steps, making it easy to understand and follow along.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at what we're dealing with. The given expression is a polynomial, which is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. In this case, the expression contains three variables: $x$ , $y$ , and $z$ , and several coefficients.

Identifying the Terms

To simplify the expression, we need to identify the individual terms. A term is a single variable or a product of variables and coefficients. In this expression, we have two terms:

$28x^2$
$-15x^5yz^2$

Understanding the Exponents

Exponents are a crucial part of algebraic expressions. In this case, we have two terms with exponents:

$x^2$ (the exponent is 2)
$x^5$ (the exponent is 5)
$y$ (no exponent, which means the exponent is 1)
$z^2$ (the exponent is 2)

Simplifying the Expression

Now that we have a good understanding of the expression, let's simplify it step by step.

Step 1: Combine Like Terms

Like terms are terms that have the same variable(s) with the same exponent(s). In this expression, we have two like terms:

$28x^2$
$-15x^5yz^2$

However, these two terms are not like terms because they have different exponents. Therefore, we cannot combine them.

Step 2: Factor Out Common Terms

Factoring out common terms involves identifying common factors in the coefficients and variables. In this expression, we can factor out a common term of $-3x^2$ from both terms:

$28x^2 = 3 \cdot 28 \cdot x^2 = 3 \cdot 4 \cdot 7 \cdot x^2$
$-15x^5yz^2 = -3 \cdot 5 \cdot x^5yz^2$

Now, we can factor out the common term of $-3x^2$ from both terms:

$28x^2 - 15x^5yz^2 = -3x^2(4 \cdot 7 - 5 \cdot x^3yz^2)$

Step 3: Simplify the Expression

Now that we have factored out the common term, we can simplify the expression further. We can rewrite the expression as:

$28x^2 - 15x^5yz^2 = -3x^2(4 \cdot 7 - 5 \cdot x^3yz^2)$

This is the simplified expression.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. In this article, we have walked through the process of simplifying the given expression: $28x^2 - 15x^5yz^2$ . We have identified the individual terms, understood the exponents, and simplified the expression step by step. By following these steps, you can simplify any algebraic expression and become more confident in your math skills.

Frequently Asked Questions

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What is a term in an algebraic expression?

A: A term is a single variable or a product of variables and coefficients.

Q: What is an exponent in an algebraic expression?

A: An exponent is a power to which a variable is raised.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to identify the individual terms, understand the exponents, and combine like terms. You can also factor out common terms to simplify the expression further.

Additional Resources

If you want to learn more about simplifying algebraic expressions, here are some additional resources:

Khan Academy: Algebraic Expressions
Mathway: Simplifying Algebraic Expressions
Wolfram Alpha: Algebraic Expression Simplifier

By following these resources and practicing your skills, you can become more confident in your ability to simplify algebraic expressions and tackle more complex math problems.

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Introduction

Simplifying algebraic expressions is a crucial skill for any math enthusiast. In our previous article, we walked through the process of simplifying the expression: $28x^2 - 15x^5yz^2$ . However, we know that math can be complex and confusing, and sometimes it's hard to understand the concepts. That's why we've put together this Q&A guide to help you tackle any algebraic expression simplification questions you may have.

Q&A: Algebraic Expression Simplification

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify the individual terms. This involves breaking down the expression into its component parts and understanding the variables and coefficients involved.

Q: How do I identify like terms in an algebraic expression?

A: Like terms are terms that have the same variable(s) with the same exponent(s). To identify like terms, look for terms that have the same base and exponent. For example, $2x^2$ and $4x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: Can I combine like terms in an algebraic expression?

A: Yes, you can combine like terms in an algebraic expression. To do this, add or subtract the coefficients of the like terms. For example, $2x^2 + 4x^2 = 6x^2$ .

Q: How do I factor out common terms in an algebraic expression?

A: To factor out common terms, identify the common factors in the coefficients and variables. Then, divide each term by the common factor. For example, $6x^2 + 12x^2 = 18x^2$ can be factored as $6x^2(1 + 2)$ .

Q: What is the difference between a term and a factor?

A: A term is a single variable or a product of variables and coefficients. A factor is a number or variable that divides another number or variable. For example, in the expression $6x^2$ , $6$ is a coefficient and $x^2$ is a term. If we factor out the common term of $2$ , we get $3x^2$ , where $3$ is a factor.

Q: Can I simplify an algebraic expression with multiple variables?

A: Yes, you can simplify an algebraic expression with multiple variables. To do this, follow the same steps as before: identify the individual terms, combine like terms, and factor out common terms.

Q: How do I simplify an algebraic expression with negative coefficients?

A: To simplify an algebraic expression with negative coefficients, follow the same steps as before. However, be careful when combining like terms, as the negative sign may affect the result.

Q: Can I simplify an algebraic expression with fractions?

A: Yes, you can simplify an algebraic expression with fractions. To do this, follow the same steps as before. However, be careful when combining like terms, as the fractions may affect the result.

Conclusion

Simplifying algebraic expressions is a crucial skill for any math enthusiast. By following the steps outlined in this Q&A guide, you can tackle any algebraic expression simplification question you may have. Remember to identify the individual terms, combine like terms, and factor out common terms. With practice and patience, you'll become more confident in your ability to simplify algebraic expressions and tackle more complex math problems.

Frequently Asked Questions

Q: What is the difference between an algebraic expression and an equation?

A: An algebraic expression is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. An equation is a statement that two expressions are equal.

Q: How do I solve an algebraic equation?

A: To solve an algebraic equation, follow the same steps as before: simplify the expression, isolate the variable, and solve for the variable.

Q: Can I simplify an algebraic expression with radicals?

A: Yes, you can simplify an algebraic expression with radicals. To do this, follow the same steps as before. However, be careful when combining like terms, as the radicals may affect the result.

Additional Resources

If you want to learn more about simplifying algebraic expressions, here are some additional resources:

Khan Academy: Algebraic Expressions
Mathway: Simplifying Algebraic Expressions
Wolfram Alpha: Algebraic Expression Simplifier

By following these resources and practicing your skills, you can become more confident in your ability to simplify algebraic expressions and tackle more complex math problems.