Analyzing the Simplified Product

Oct 13, 2025 by windailys.com 62 views

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Understanding the Problem

In this article, we will delve into the world of algebraic expressions and simplify a given product. The problem at hand is to determine the truth value of certain statements about the fully simplified product of $(b - 2c)(-3b + c)$ . We will break down the solution step by step, using algebraic techniques to simplify the expression and identify the number of terms, degree, and other relevant characteristics.

Simplifying the Product

To simplify the product $(b - 2c)(-3b + c)$ , we will use the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . We will apply this property to each term in the first expression and then multiply the resulting terms.

(b - 2c)(-3b + c) = b(-3b + c) - 2c(-3b + c)

Expanding the Terms

Now, we will expand each term using the distributive property.

b(-3b + c) = -3b^2 + bc
-2c(-3b + c) = 6bc - 2c^2

Combining Like Terms

Next, we will combine like terms by adding or subtracting the coefficients of the same variables.

-3b^2 + bc + 6bc - 2c^2 = -3b^2 + 7bc - 2c^2

Analyzing the Simplified Product

Now that we have simplified the product, we can analyze its characteristics. The simplified product has 3 terms: $-3b^2$ , $7bc$ , and $-2c^2$ . This means that statement A, "The simplified product has 2 terms," is false. On the other hand, statement B, "The simplified product has 4 terms," is also false. However, statement C, "The simplified product has a degree of 2," is true, as the highest power of any variable in the simplified product is 2.

Conclusion

In conclusion, the fully simplified product of $(b - 2c)(-3b + c)$ is $-3b^2 + 7bc - 2c^2$ . This expression has 3 terms and a degree of 2. Therefore, statement C is true, while statements A and B are false.

Final Answer

Based on our analysis, the correct answer is:

C. The simplified product has a degree of 2.

Note that statement D is not relevant to the problem and is therefore not considered.

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Understanding Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will address some of the most frequently asked questions about simplifying algebraic expressions.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . This property allows us to expand expressions by multiplying each term inside the parentheses by the factor outside the parentheses.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to follow these steps:

Distribute: Use the distributive property to expand the expression by multiplying each term inside the parentheses by the factor outside the parentheses.
Combine like terms: Add or subtract the coefficients of the same variables to simplify the expression.
Simplify: Simplify the expression by combining like terms and eliminating any unnecessary terms.

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

A: A term is a single part of an algebraic expression, such as $2x$ or $3y$ . A factor is a number or variable that is multiplied together to form a term. For example, in the expression $2x(3y)$ , the factors are $2$ , $x$ , $3$ , and $y$ .

Q: How do I identify like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1. On the other hand, $2x$ and $3y$ are not like terms because they have different variables.

Q: Can I simplify an expression with variables raised to different powers?

A: Yes, you can simplify an expression with variables raised to different powers. However, you need to be careful when combining like terms. For example, in the expression $2x^2 + 3x$ , you can combine the like terms $2x^2$ and $3x$ to get $5x^2 + 3x$ .

Q: What is the degree of an algebraic expression?

A: The degree of an algebraic expression is the highest power of any variable in the expression. For example, in the expression $2x^2 + 3x$ , the degree is 2 because the highest power of the variable $x$ is 2.

Q: Can I simplify an expression with negative coefficients?

A: Yes, you can simplify an expression with negative coefficients. When simplifying an expression with negative coefficients, you need to be careful when combining like terms. For example, in the expression $-2x + 3x$ , you can combine the like terms to get $x$ .

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill to master in mathematics. By understanding the distributive property, combining like terms, and simplifying expressions, you can simplify even the most complex algebraic expressions. We hope this article has helped you to better understand the basics of simplifying algebraic expressions.

Final Tips

Always follow the order of operations (PEMDAS) when simplifying algebraic expressions.
Use the distributive property to expand expressions.
Combine like terms to simplify expressions.
Simplify expressions by eliminating any unnecessary terms.
Practice, practice, practice! The more you practice simplifying algebraic expressions, the more comfortable you will become with the process.