Simplify $\left(x^2+16\right)\left(x^2-16\right$\].A. $x^4+256$ B. $x^4+32$ C. $x^4-32$ D. $x^4-256$

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Introduction

In this article, we will simplify the given expression $\left(x^2+16\right)\left(x^2-16\right)$ using the difference of squares formula. The difference of squares formula is a fundamental concept in algebra that allows us to simplify expressions of the form $(a+b)(a-b)$.

The Difference of Squares Formula

The difference of squares formula states that for any two numbers $a$ and $b$, the product of their sum and difference is equal to the difference of their squares:

$$\left(a+b\right)\left(a-b\right)=a^2-b^2$$

Applying the Difference of Squares Formula

We can apply the difference of squares formula to the given expression $\left(x^2+16\right)\left(x^2-16\right)$ by letting $a=x^2$ and $b=16$. Then, we have:

$$\left(x^2+16\right)\left(x^2-16\right)=\left(x^2\right)^2-\left(16\right)^2$$

Simplifying the Expression

Now, we can simplify the expression by evaluating the squares:

$$\left(x^2\right)^2-\left(16\right)^2=x^4-256$$

Conclusion

Therefore, the simplified expression is $x^4-256$. This is the correct answer.

Comparison with Other Options

Let's compare our answer with the other options:

• Option A: $x^4+256$ is incorrect because the expression is a difference of squares, not a sum.
• Option B: $x^4+32$ is incorrect because the expression is a difference of squares, not a sum, and the constant term is incorrect.
• Option C: $x^4-32$ is incorrect because the constant term is incorrect.

Final Answer

The final answer is $\boxed{x^4-256}$.

Frequently Asked Questions

Q: What is the difference of squares formula?

A: The difference of squares formula is a fundamental concept in algebra that allows us to simplify expressions of the form $(a+b)(a-b)$.

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, let $a$ be the first term and $b$ be the second term. Then, the product of their sum and difference is equal to the difference of their squares.

Q: What is the simplified expression?

A: The simplified expression is $x^4-256$.

Q: Why is option A incorrect?

A: Option A is incorrect because the expression is a difference of squares, not a sum.

Q: Why is option B incorrect?

A: Option B is incorrect because the expression is a difference of squares, not a sum, and the constant term is incorrect.

Q: Why is option C incorrect?

A: Option C is incorrect because the constant term is incorrect.

Step-by-Step Solution

1. Let $a=x^2$ and $b=16$.
2. Apply the difference of squares formula: $\left(x^2+16\right)\left(x^2-16\right)=\left(x^2\right)^2-\left(16\right)^2$.
3. Simplify the expression: $x^4-256$.

Common Mistakes

• Forgetting to apply the difference of squares formula.
• Not simplifying the expression correctly.
• Choosing the wrong option.

Tips and Tricks

• Make sure to apply the difference of squares formula correctly.
• Simplify the expression carefully.
• Choose the correct option based on the simplified expression.

Real-World Applications

• The difference of squares formula has many real-world applications, such as in physics and engineering.
• It can be used to simplify complex expressions and solve problems.

Conclusion

In conclusion, the simplified expression is $x^4-256$. This is the correct answer. The difference of squares formula is a fundamental concept in algebra that allows us to simplify expressions of the form $(a+b)(a-b)$. By applying the difference of squares formula and simplifying the expression, we can arrive at the correct answer.

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Introduction

In our previous article, we simplified the expression $\left(x^2+16\right)\left(x^2-16\right)$ using the difference of squares formula. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the difference of squares formula?

A: The difference of squares formula is a fundamental concept in algebra that allows us to simplify expressions of the form $(a+b)(a-b)$. It states that for any two numbers $a$ and $b$, the product of their sum and difference is equal to the difference of their squares:

$$\left(a+b\right)\left(a-b\right)=a^2-b^2$$

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, let $a$ be the first term and $b$ be the second term. Then, the product of their sum and difference is equal to the difference of their squares.

Q: What is the simplified expression?

A: The simplified expression is $x^4-256$.

Q: Why is option A incorrect?

A: Option A is incorrect because the expression is a difference of squares, not a sum.

Q: Why is option B incorrect?

A: Option B is incorrect because the expression is a difference of squares, not a sum, and the constant term is incorrect.

Q: Why is option C incorrect?

A: Option C is incorrect because the constant term is incorrect.

Q: Can I use the difference of squares formula for other types of expressions?

A: Yes, you can use the difference of squares formula for other types of expressions, such as $(a-b)(a-b)$ or $(a+b)(a+b)$.

Q: How do I know which option is correct?

A: To determine which option is correct, simplify the expression using the difference of squares formula and compare it with the options.

Q: What are some common mistakes to avoid?

A: Some common mistakes to avoid include forgetting to apply the difference of squares formula, not simplifying the expression correctly, and choosing the wrong option.

Q: What are some tips and tricks for simplifying expressions?

A: Some tips and tricks for simplifying expressions include making sure to apply the difference of squares formula correctly, simplifying the expression carefully, and choosing the correct option based on the simplified expression.

Q: How do I apply the difference of squares formula in real-world applications?

A: The difference of squares formula has many real-world applications, such as in physics and engineering. It can be used to simplify complex expressions and solve problems.

Step-by-Step Solution

1. Let $a=x^2$ and $b=16$.
2. Apply the difference of squares formula: $\left(x^2+16\right)\left(x^2-16\right)=\left(x^2\right)^2-\left(16\right)^2$.
3. Simplify the expression: $x^4-256$.

Common Mistakes

• Forgetting to apply the difference of squares formula.
• Not simplifying the expression correctly.
• Choosing the wrong option.

Tips and Tricks

• Make sure to apply the difference of squares formula correctly.
• Simplify the expression carefully.
• Choose the correct option based on the simplified expression.

Real-World Applications

• The difference of squares formula has many real-world applications, such as in physics and engineering.
• It can be used to simplify complex expressions and solve problems.

Conclusion

In conclusion, the simplified expression is $x^4-256$. This is the correct answer. The difference of squares formula is a fundamental concept in algebra that allows us to simplify expressions of the form $(a+b)(a-b)$. By applying the difference of squares formula and simplifying the expression, we can arrive at the correct answer.

Additional Resources

• For more information on the difference of squares formula, visit our website or consult a math textbook.
• Practice simplifying expressions using the difference of squares formula with our online exercises.
• Watch video tutorials on the difference of squares formula on YouTube or other online platforms.

Final Answer

The final answer is $\boxed{x^4-256}$.