How To Simplify The Expression $3x^2(7x^2-9)$ A Step By Step Guide

In mathematics, simplifying algebraic expressions is a fundamental skill. It involves reducing an expression to its most basic and understandable form without changing its mathematical value. This often makes it easier to solve equations, understand relationships, and perform further calculations. In this comprehensive guide, we will delve into the simplification of the expression $3x^2(7x^2-9)$, exploring the underlying principles and techniques involved. We will break down the steps, explain the mathematical rationale behind each one, and highlight common pitfalls to avoid. By the end of this guide, you will have a solid understanding of how to simplify similar expressions and apply these skills to more complex algebraic problems.

Understanding the Distributive Property

At the heart of simplifying the expression $3x^2(7x^2-9)$ lies the distributive property. The distributive property is a fundamental concept in algebra that allows us to multiply a single term by an expression enclosed in parentheses. In its most basic form, the distributive property states that for any numbers a, b, and c:

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

This means that the term outside the parentheses (a) is multiplied by each term inside the parentheses (b and c) individually. The results are then added together. This property extends to expressions with subtraction as well:

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

In our case, the expression $3x^2(7x^2-9)$ fits this pattern perfectly. We have a single term, $3x^2$, multiplying an expression in parentheses, $(7x^2-9)$. To simplify this, we need to distribute $3x^2$ to both terms inside the parentheses, $7x^2$ and $-9$. Understanding this foundational principle is crucial for correctly simplifying the expression. Misapplication of the distributive property is a common source of errors in algebra, so it's essential to grasp its mechanics thoroughly. We will see how this property is applied step-by-step in the following sections, ensuring a clear and accurate simplification process.

Step-by-Step Simplification of $3x^2(7x^2-9)$

To simplify the expression $3x^2(7x^2-9)$, we will meticulously apply the distributive property, ensuring each step is clear and mathematically sound. Here’s a detailed breakdown:

1. Applying the Distributive Property: As discussed earlier, the distributive property is the key to simplifying this expression. We need to multiply the term outside the parentheses, $3x^2$, by each term inside the parentheses, $7x^2$ and $-9$. This gives us:

   $3x^2 * (7x^2) - 3x^2 * (9)$

   Notice how the minus sign between the terms inside the parentheses is preserved. This is crucial for maintaining the correct mathematical relationship within the expression. Now we have two separate multiplication operations to perform.

2. Multiplying the First Terms: Let's focus on the first multiplication: $3x^2 * 7x^2$. To multiply these terms, we multiply the coefficients (the numerical parts) and then multiply the variables. The coefficients are 3 and 7, and their product is $3 * 7 = 21$. For the variables, we have $x^2 * x^2$. When multiplying variables with exponents, we add the exponents. In this case, $2 + 2 = 4$, so $x^2 * x^2 = x^4$. Therefore, the product of the first terms is:

   $3x^2 * 7x^2 = 21x^4$

   This step combines numerical multiplication with the rules of exponents, a fundamental skill in algebra.

3. Multiplying the Second Terms: Now let's tackle the second multiplication: $3x^2 * 9$. Again, we multiply the coefficients first: $3 * 9 = 27$. The variable part is simply $x^2$ since there is no variable term in 9. Thus, the product is:

   $3x^2 * 9 = 27x^2$

4. Combining the Results: Now that we have the results of both multiplications, we can combine them back into a single expression. Remember that we had a minus sign between the two products from the distributive property. So, we subtract the second term from the first term:

   $21x^4 - 27x^2$

   This is the simplified form of the original expression. We have successfully distributed, multiplied, and combined the terms to arrive at the final answer.

Identifying the Correct Answer

Given the options:

• A. $21x^2 - 9$
• B. $21x^2 - 27x^2$
• C. $21x^4 - 9$
• D. $21x^4 - 27x^2$

Our simplified expression, derived in the previous section, is $21x^4 - 27x^2$. Comparing this result with the given options, we can clearly see that:

• Option A, $21x^2 - 9$, is incorrect because it does not have the correct exponents for the variables. It also fails to account for the multiplication of $3x^2$ by $7x^2$ and $9$ accurately.
• Option B, $21x^2 - 27x^2$, is incorrect as well. While it includes the $27x^2$ term, it incorrectly states the first term as $21x^2$ instead of $21x^4$. This error arises from not correctly applying the exponent rules during multiplication.
• Option C, $21x^4 - 9$, is also incorrect. It correctly identifies the $21x^4$ term but fails to multiply $3x^2$ by the second term, $-9$, inside the parentheses. This shows a misunderstanding of the distributive property's full application.
• Option D, $21x^4 - 27x^2$, is the correct answer. This option matches our simplified expression exactly. It correctly applies the distributive property, multiplies the terms, and combines the results with the correct signs and exponents.

Therefore, after a careful step-by-step simplification and comparison with the provided options, we can confidently identify D. $21x^4 - 27x^2$ as the correct answer.

Common Mistakes to Avoid

Simplifying algebraic expressions can be tricky, and there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate simplification. Here are some common mistakes to watch out for when simplifying expressions like $3x^2(7x^2-9)$:

1. Incorrectly Applying the Distributive Property:

   • Mistake: Forgetting to multiply the term outside the parentheses by every term inside. For example, multiplying $3x^2$ by $7x^2$ but not by $-9$. This leads to an incomplete simplification.
   • How to Avoid: Always ensure that the term outside the parentheses is distributed to each term inside. Draw arrows or write out each multiplication step explicitly to keep track.

2. Errors with Exponent Rules:

   • Mistake: Incorrectly adding exponents when multiplying terms with the same base. For instance, stating $x^2 * x^2 = x^3$ instead of the correct $x^4$.
   • How to Avoid: Remember the rule: when multiplying variables with the same base, add the exponents ($x^m * x^n = x^{m+n}$). Review exponent rules regularly and practice applying them.

3. Sign Errors:

   • Mistake: Not paying close attention to the signs (positive or negative) of the terms. For example, incorrectly distributing $3x^2$ to $-9$ and writing $+27x^2$ instead of $-27x^2$.
   • How to Avoid: Be meticulous with signs. When distributing, pay attention to whether you are multiplying by a positive or negative term. Write out the signs explicitly in each step.

4. Combining Unlike Terms:

   • Mistake: Adding or subtracting terms that do not have the same variable and exponent. For example, incorrectly combining $21x^4$ and $-27x^2$.
   • How to Avoid: Remember that terms can only be combined if they are “like terms,” meaning they have the same variable raised to the same power. Identify like terms before attempting to combine them.

5. Order of Operations:

   • Mistake: Not following the correct order of operations (PEMDAS/BODMAS). While less relevant in this specific example, it's crucial in more complex expressions.
   • How to Avoid: Always follow the order of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

By being mindful of these common mistakes and actively working to avoid them, you can significantly improve your accuracy in simplifying algebraic expressions.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, practice is essential. Here are a few practice problems similar to the one we've worked through, allowing you to apply the distributive property and exponent rules. Try solving these on your own, and then compare your answers to the solutions provided.

Practice Problems:

1. Simplify: $2x^3(5x^2 - 4)$
2. Simplify: $-4y^2(3y^3 + 2y)$
3. Simplify: $x(8x^4 - 6x^2 + 1)$

Solutions:

1. Solution: $2x^3(5x^2 - 4)$

   • Distribute $2x^3$ to both terms inside the parentheses: $2x^3 * 5x^2 - 2x^3 * 4$
   • Multiply the coefficients and add the exponents: $10x^5 - 8x^3$

   Therefore, the simplified expression is $10x^5 - 8x^3$.

2. Solution: $-4y^2(3y^3 + 2y)$

   • Distribute $-4y^2$ to both terms inside the parentheses: $-4y^2 * 3y^3 + (-4y^2) * 2y$
   • Multiply the coefficients and add the exponents, paying attention to the signs: $-12y^5 - 8y^3$

   Thus, the simplified expression is $-12y^5 - 8y^3$.

3. Solution: $x(8x^4 - 6x^2 + 1)$

   • Distribute $x$ to each term inside the parentheses: $x * 8x^4 - x * 6x^2 + x * 1$
   • Multiply the coefficients and add the exponents: $8x^5 - 6x^3 + x$

   Hence, the simplified expression is $8x^5 - 6x^3 + x$.

By working through these practice problems, you reinforce your understanding of the simplification process and become more confident in applying the distributive property and exponent rules. Remember to always double-check your work and be mindful of the common mistakes discussed earlier. Continued practice will help you master these essential algebraic skills.

Conclusion

In conclusion, simplifying algebraic expressions is a vital skill in mathematics, and understanding the distributive property is key to success. By meticulously applying the steps we’ve outlined, you can confidently simplify expressions like $3x^2(7x^2-9)$ and similar problems. Remember to distribute correctly, pay attention to exponent rules and signs, and avoid combining unlike terms. The correct answer to our initial problem is D. $21x^4 - 27x^2$. Practice is crucial for mastering these concepts, so work through the provided examples and seek out additional problems to build your proficiency. With a solid grasp of these fundamental techniques, you’ll be well-equipped to tackle more complex algebraic challenges. Keep practicing, and you’ll see your skills and confidence grow.