Distributive Property and Combining Like Terms

Oct 13, 2025 by miodeya.com 49 views

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Understanding the Problem

When subtracting polynomials, it's essential to remember that we're dealing with the difference between two algebraic expressions. In this case, we're given the expression $\left(-3x^4 - 12\right) - 2x^4$ , and we need to simplify it to its simplest terms.

Distributive Property and Combining Like Terms

To simplify the given expression, we'll first apply the distributive property to remove the parentheses. This means we'll multiply the negative sign outside the parentheses by each term inside. The expression becomes:

-3x^4 - 12 - 2x^4

Now, we can combine like terms, which involves adding or subtracting terms that have the same variable and exponent. In this case, we have two terms with the variable $x^4$ . We can combine them by adding their coefficients:

-3x^4 - 2x^4 = (-3 - 2)x^4 = -5x^4

Combining the Constant Terms

Next, we'll combine the constant terms, which are the terms without any variables. In this case, we have the terms $-12$ and $0$ (since there are no other constant terms). We can combine them by adding their values:

-12 + 0 = -12

Simplifying the Expression

Now that we've combined like terms and constant terms, we can simplify the expression by combining the results:

-5x^4 - 12

Final Answer

The final answer is $\boxed{-5x^4 - 12}$ .

Conclusion

Subtracting polynomials involves applying the distributive property and combining like terms. By following these steps, we can simplify complex algebraic expressions and arrive at their simplest forms. In this case, we started with the expression $\left(-3x^4 - 12\right) - 2x^4$ and simplified it to $-5x^4 - 12$ .

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Q: What is the distributive property in the context of subtracting polynomials?

A: The distributive property is a mathematical concept that allows us to remove parentheses by multiplying the negative sign outside the parentheses by each term inside. In the context of subtracting polynomials, it helps us to simplify the expression by distributing the negative sign to each term.

Q: How do I combine like terms when subtracting polynomials?

A: To combine like terms, you need to add or subtract terms that have the same variable and exponent. For example, if you have two terms with the variable $x^4$ , you can combine them by adding their coefficients. In the expression $\left(-3x^4 - 12\right) - 2x^4$ , we combined the two terms with the variable $x^4$ by adding their coefficients: $-3x^4 - 2x^4 = (-3 - 2)x^4 = -5x^4$ .

Q: What are constant terms in the context of subtracting polynomials?

A: Constant terms are terms without any variables. In the expression $\left(-3x^4 - 12\right) - 2x^4$ , the constant terms are $-12$ and $0$ (since there are no other constant terms). We combine constant terms by adding their values.

Q: Can I simplify the expression by combining the constant terms first?

A: Yes, you can simplify the expression by combining the constant terms first. However, it's often more efficient to combine like terms first, as this can help you simplify the expression more quickly.

Q: What is the final answer to the expression $\left(-3x^4 - 12\right) - 2x^4$ ?

A: The final answer to the expression $\left(-3x^4 - 12\right) - 2x^4$ is $\boxed{-5x^4 - 12}$ .

Q: Can I use the distributive property to simplify expressions with multiple variables?

A: Yes, you can use the distributive property to simplify expressions with multiple variables. However, you need to be careful to distribute the negative sign to each term correctly.

Q: What are some common mistakes to avoid when subtracting polynomials?

A: Some common mistakes to avoid when subtracting polynomials include:

Failing to distribute the negative sign correctly
Not combining like terms correctly
Forgetting to combine constant terms
Not simplifying the expression to its simplest form

Q: How can I practice subtracting polynomials?

A: You can practice subtracting polynomials by working through examples and exercises in your textbook or online resources. You can also try creating your own examples and simplifying them to test your understanding.

Q: What are some real-world applications of subtracting polynomials?

A: Subtracting polynomials has many real-world applications, including:

Simplifying algebraic expressions in physics and engineering
Solving systems of equations in economics and finance
Modeling population growth and decay in biology and ecology
Analyzing data in statistics and data science

Q: Can I use technology to help me simplify expressions when subtracting polynomials?

A: Yes, you can use technology to help you simplify expressions when subtracting polynomials. Many graphing calculators and computer algebra systems (CAS) can simplify expressions and perform calculations quickly and accurately.