Simplify: $\[ -3x^3(-2x^2 + 4x - 3) \\]Options: A. \[$6x^5 - 12x^4 + 9x^3\$\] B. \[$6x^6 - 12x^3 + 9\$\] C. \[$-5x^5 + X^4 - 6x^3\$\] D. \[$-6x^6 - 12x^4 + 9x^3\$\]

Understanding the Problem

The given problem involves simplifying an algebraic expression using the distributive property. The expression to be simplified is $-3x^3(-2x^2 + 4x - 3)$. To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Step 1: Apply the Distributive Property

To simplify the given expression, we need to apply the distributive property. We can start by multiplying the term $-3x^3$ with each term inside the parentheses.

$$-3x^3(-2x^2 + 4x - 3) = -3x^3(-2x^2) + -3x^3(4x) + -3x^3(-3)$$

Step 2: Simplify Each Term

Now, we need to simplify each term by multiplying the coefficients and adding the exponents of the variables.

$$-3x^3(-2x^2) = 6x^5$$

$$-3x^3(4x) = -12x^4$$

$$-3x^3(-3) = 9x^3$$

Step 3: Combine the Terms

Now, we can combine the simplified terms to get the final expression.

$$6x^5 - 12x^4 + 9x^3$$

Comparing with the Options

Now, we need to compare the simplified expression with the given options to find the correct answer.

Conclusion

Based on the simplification, we can see that the correct answer is option A, which is $6x^5 - 12x^4 + 9x^3$.

Key Takeaways

• The distributive property is a fundamental concept in algebra that allows us to simplify complex expressions.
• To simplify an expression using the distributive property, we need to multiply each term inside the parentheses with the term outside the parentheses.
• We need to be careful when multiplying variables with exponents, as we need to add the exponents when multiplying like bases.

Practice Problems

1. Simplify the expression $2x^2(3x - 2)$.
2. Simplify the expression $-4x^3(2x^2 + 3x - 1)$.
3. Simplify the expression $x^2(2x + 3)$.

Answer Key

1. $6x^3 - 4x^2$
2. $-8x^5 - 12x^4 + 4x^3$
3. $2x^3 + 3x^2$

Additional Resources

• Khan Academy: Algebra
• Mathway: Algebra Calculator
• Wolfram Alpha: Algebra Solver
  Simplify: $-3x^3(-2x^2 + 4x - 3)$ Q&A =====================================================

Q: What is the distributive property in algebra?

A: The distributive property is a fundamental concept in algebra that allows us to simplify complex expressions. It states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This means that we can multiply a single term with each term inside the parentheses.

Q: How do I apply the distributive property to simplify an expression?

A: To apply the distributive property, we need to multiply each term inside the parentheses with the term outside the parentheses. We can start by multiplying the term outside the parentheses with each term inside the parentheses, and then simplify each term by multiplying the coefficients and adding the exponents of the variables.

Q: What is the correct order of operations when simplifying an expression using the distributive property?

A: The correct order of operations is:

1. Multiply each term inside the parentheses with the term outside the parentheses.
2. Simplify each term by multiplying the coefficients and adding the exponents of the variables.
3. Combine the simplified terms to get the final expression.

Q: How do I handle negative coefficients when simplifying an expression using the distributive property?

A: When simplifying an expression using the distributive property, we need to handle negative coefficients carefully. A negative coefficient can be moved to the other side of the expression by changing its sign. For example, if we have the expression $-3x^3(-2x^2 + 4x - 3)$, we can move the negative coefficient $-3x^3$ to the other side of the expression by changing its sign to $3x^3$.

Q: What are some common mistakes to avoid when simplifying expressions using the distributive property?

A: Some common mistakes to avoid when simplifying expressions using the distributive property include:

• Forgetting to multiply each term inside the parentheses with the term outside the parentheses.
• Not simplifying each term by multiplying the coefficients and adding the exponents of the variables.
• Not combining the simplified terms to get the final expression.

Q: How can I practice simplifying expressions using the distributive property?

A: You can practice simplifying expressions using the distributive property by working through practice problems. Some examples of practice problems include:

• Simplifying the expression $2x^2(3x - 2)$.
• Simplifying the expression $-4x^3(2x^2 + 3x - 1)$.
• Simplifying the expression $x^2(2x + 3)$.

Q: What are some real-world applications of the distributive property in algebra?

A: The distributive property has many real-world applications in algebra, including:

• Simplifying complex expressions in physics and engineering.
• Solving systems of linear equations in economics and finance.
• Modeling population growth and decay in biology and medicine.

Q: How can I use technology to help me simplify expressions using the distributive property?

A: You can use technology, such as algebra calculators or computer algebra systems, to help you simplify expressions using the distributive property. These tools can help you to:

• Simplify complex expressions quickly and accurately.
• Check your work for errors.
• Explore different solutions to a problem.

Conclusion

In conclusion, the distributive property is a fundamental concept in algebra that allows us to simplify complex expressions. By understanding how to apply the distributive property, we can simplify expressions and solve problems in a variety of fields. With practice and patience, you can become proficient in simplifying expressions using the distributive property.