Distributive Property and Like Terms

Oct 13, 2025 by miodeya.com 141 views

Understanding the Problem

The given problem involves simplifying an algebraic expression by combining like terms. The expression to be simplified is $\left(-3x^2 + x + 5\right) - \left(4x^2 - 2x\right)$ . To simplify this expression, we need to apply the rules of algebra, specifically the distributive property and the concept of like terms.

Distributive Property and Like Terms

The distributive property states that for any real numbers a, b, and c, the following equation holds: $a(b + c) = ab + ac$ . This property allows us to distribute a single term across the terms inside the parentheses. In the given expression, we can apply the distributive property to remove the parentheses and simplify the expression.

Step 1: Apply the Distributive Property

To simplify the given expression, we can start by applying the distributive property to remove the parentheses. This will allow us to combine like terms and simplify the expression.

$\left(-3x^2 + x + 5\right) - \left(4x^2 - 2x\right)$

Using the distributive property, we can rewrite the expression as:

$-3x^2 + x + 5 - 4x^2 + 2x$

Step 2: Combine Like Terms

Now that we have removed the parentheses, we can combine like terms to simplify the expression. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable $x^2$ and two terms with the variable $x$ .

$-3x^2 + x + 5 - 4x^2 + 2x$

Combining the like terms, we get:

$-7x^2 + 3x + 5$

Conclusion

The simplified form of the given expression is $-7x^2 + 3x + 5$ . This is the final answer to the problem.

Comparison with Answer Choices

Now that we have simplified the expression, we can compare our answer with the answer choices provided. The correct answer is:

D. $-7x^2 + 3x + 5$

This answer choice matches our simplified expression, confirming that it is the correct solution to the problem.

Final Answer

The final answer to the problem is:

D. $-7x^2 + 3x + 5$

This answer choice represents the simplified form of the given expression, and it is the correct solution to the problem.

Frequently Asked Questions

Q: What is the distributive property in algebra?

Q: What are like terms in algebra?

A: Like terms are terms that have the same variable raised to the same power. In the given expression, we have two terms with the variable $x^2$ and two terms with the variable $x$ .

Q: How do we combine like terms in algebra?

A: To combine like terms, we add or subtract the coefficients of the like terms. In the given expression, we combined the like terms by adding the coefficients of the $x^2$ terms and the $x$ terms.

Additional Resources

For more information on algebra and simplifying expressions, you can refer to the following resources:

Khan Academy: Algebra
Mathway: Algebra Calculator
Wolfram Alpha: Algebra Solver

These resources provide additional information and practice problems to help you improve your skills in algebra and simplifying expressions.

Q&A: Simplifying Algebraic Expressions

Q: What is the distributive property in algebra?

A: The distributive property is a rule in algebra that allows us to distribute a single term across the terms inside the parentheses. It states that for any real numbers a, b, and c, the following equation holds: $a(b + c) = ab + ac$ . This property is essential in simplifying algebraic expressions by removing parentheses and combining like terms.

Q: What are like terms in algebra?

A: Like terms are terms that have the same variable raised to the same power. In the given expression, we have two terms with the variable $x^2$ and two terms with the variable $x$ . Like terms can be combined by adding or subtracting their coefficients.

Q: How do we combine like terms in algebra?

A: To combine like terms, we add or subtract the coefficients of the like terms. In the given expression, we combined the like terms by adding the coefficients of the $x^2$ terms and the $x$ terms. For example, if we have two terms with the variable $x$ , we can combine them as follows:

$x + 3x = 4x$

Q: What is the order of operations in algebra?

A: The order of operations in algebra is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is as follows:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do we simplify expressions with negative coefficients?

A: When simplifying expressions with negative coefficients, we need to remember that a negative coefficient is equivalent to a positive coefficient with a negative sign. For example, if we have the expression $-3x^2 + x + 5$ , we can simplify it by combining like terms:

$-3x^2 + x + 5 = -7x^2 + 3x + 5$

Q: What is the difference between a variable and a constant in algebra?

A: In algebra, a variable is a letter or symbol that represents a value that can change. A constant, on the other hand, is a value that does not change. For example, in the expression $2x + 5$ , the variable is $x$ and the constant is $5$ .

Q: How do we simplify expressions with fractions?

A: When simplifying expressions with fractions, we need to remember that a fraction is a way of representing a part of a whole. To simplify an expression with fractions, we can multiply both the numerator and the denominator by the same value to eliminate the fraction. For example, if we have the expression $\frac{1}{2}x + \frac{1}{2}x$ , we can simplify it by combining like terms:

$\frac{1}{2}x + \frac{1}{2}x = x$

Additional Resources

For more information on algebra and simplifying expressions, you can refer to the following resources:

Khan Academy: Algebra
Mathway: Algebra Calculator
Wolfram Alpha: Algebra Solver

These resources provide additional information and practice problems to help you improve your skills in algebra and simplifying expressions.

Practice Problems

Try simplifying the following expressions:

$2x^2 + 3x + 5 - 4x^2 + 2x$
$-3x^2 + 2x + 5 + 4x^2 - 2x$
$\frac{1}{2}x + \frac{1}{2}x$
$2x^2 + 3x + 5 - 2x^2 + 3x$

Answers:

$-2x^2 + 5x + 5$
$x^2 + 5$
$x$
$5x + 5$

Conclusion

Simplifying algebraic expressions is an essential skill in algebra that requires a deep understanding of the distributive property, like terms, and the order of operations. By following the steps outlined in this article, you can simplify even the most complex expressions and become proficient in algebra.