Simplify The Expression: − 5 X 4 ( − 3 X 2 + 4 X − 2 ) -5 X^4\left(-3 X^2+4 X-2\right) − 5 X 4 ( − 3 X 2 + 4 X − 2 ) A. 15 X 6 − 20 X 5 + 10 X 4 15 X^6-20 X^5+10 X^4 15 X 6 − 20 X 5 + 10 X 4 B. 15 X 8 − 20 X 4 + 10 15 X^8-20 X^4+10 15 X 8 − 20 X 4 + 10 C. − 15 X 8 − 20 X 5 + 10 X 4 -15 X^8-20 X^5+10 X^4 − 15 X 8 − 20 X 5 + 10 X 4 D. − 8 X 6 − X 5 − 7 X 4 -8 X^6-x^5-7 X^4 − 8 X 6 − X 5 − 7 X 4

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. In this article, we will focus on simplifying a specific expression involving exponents and polynomials. We will break down the expression step by step, using the distributive property and exponent rules to arrive at the final simplified form.

The Expression to Simplify

The given expression is:

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

Our goal is to simplify this expression by applying the distributive property and exponent rules.

Step 1: Apply the Distributive Property

The distributive property states that for any real numbers a, b, and c:

$$a(b+c) = ab + ac$$

We can apply this property to the given expression by distributing the $-5 x^4$ term to each term inside the parentheses:

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

Step 2: Simplify Each Term

Now, we can simplify each term by applying the exponent rules:

• $-5 x^4(-3 x^2) = 15 x^6$
• $-5 x^4(4 x) = -20 x^5$
• $-5 x^4(-2) = 10 x^4$

So, the expression becomes:

$$15 x^6 - 20 x^5 + 10 x^4$$

Conclusion

By applying the distributive property and exponent rules, we have simplified the given expression to:

$$15 x^6 - 20 x^5 + 10 x^4$$

This is the final answer.

Comparison with Answer Choices

Let's compare our simplified expression with the answer choices:

• A. $15 x^6-20 x^5+10 x^4$ (matches our simplified expression)
• B. $15 x^8-20 x^4+10$ (does not match)
• C. $-15 x^8-20 x^5+10 x^4$ (does not match)
• D. $-8 x^6-x^5-7 x^4$ (does not match)

Our simplified expression matches answer choice A.

Tips and Tricks

When simplifying expressions involving exponents and polynomials, remember to:

• Apply the distributive property to distribute terms to each term inside the parentheses
• Simplify each term by applying exponent rules
• Check your work by comparing your simplified expression with the answer choices

By following these tips and tricks, you can simplify expressions like this one with ease.

Common Mistakes to Avoid

When simplifying expressions, be careful not to:

• Forget to apply the distributive property
• Simplify each term incorrectly
• Compare your simplified expression with the answer choices incorrectly

By avoiding these common mistakes, you can ensure that your simplified expression is accurate and correct.

Real-World Applications

Simplifying expressions like this one has real-world applications in various fields, such as:

• Physics: Simplifying expressions is crucial in solving equations that describe the motion of objects.
• Engineering: Simplifying expressions is essential in designing and analyzing complex systems.
• Computer Science: Simplifying expressions is necessary in writing efficient algorithms and data structures.

By mastering the art of simplifying expressions, you can apply this skill to real-world problems and make a meaningful impact in your chosen field.

Conclusion

Q&A: Simplifying Expressions

Q: What is the distributive property, and how is it used in simplifying expressions?

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 is used to simplify expressions by distributing a term to each term inside the parentheses.

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

A: To apply the distributive property, simply multiply the term outside the parentheses to each term inside the parentheses. For example, if we have the expression:

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

We can apply the distributive property by distributing the $-5 x^4$ term to each term inside the parentheses:

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

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Forgetting to apply the distributive property
• Simplifying each term incorrectly
• Comparing your simplified expression with the answer choices incorrectly

Q: How do I check my work when simplifying an expression?

A: To check your work, simply compare your simplified expression with the answer choices. Make sure that your simplified expression matches one of the answer choices.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has real-world applications in various fields, such as:

• Physics: Simplifying expressions is crucial in solving equations that describe the motion of objects.
• Engineering: Simplifying expressions is essential in designing and analyzing complex systems.
• Computer Science: Simplifying expressions is necessary in writing efficient algorithms and data structures.

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through practice problems, such as those found in algebra textbooks or online resources. You can also try simplifying expressions on your own, using real-world examples or scenarios.

Q: What are some tips for mastering the art of simplifying expressions?

A: Some tips for mastering the art of simplifying expressions include:

• Practicing regularly to build your skills and confidence
• Using real-world examples or scenarios to make the concept more meaningful
• Checking your work carefully to ensure accuracy

Q: How can I apply the concept of simplifying expressions to real-world problems?

A: You can apply the concept of simplifying expressions to real-world problems by using the distributive property and exponent rules to simplify complex expressions. This can be useful in a variety of fields, such as physics, engineering, and computer science.

Conclusion

In conclusion, simplifying expressions is a crucial skill that has real-world applications in various fields. By mastering the art of simplifying expressions, you can apply this skill to real-world problems and make a meaningful impact in your chosen field. Remember to practice regularly, use real-world examples or scenarios, and check your work carefully to ensure accuracy.