Which Expression Is Equivalent To ( 4 X 5 + 11 ) 2 \left(4 X^5+11\right)^2 ( 4 X 5 + 11 ) 2 ?A. 16 X 10 + 121 16 X^{10} + 121 16 X 10 + 121 B. 16 X 40 + 121 16 X^{40} + 121 16 X 40 + 121 C. 16 X 10 + 88 X 5 + 121 16 X^{10} + 88 X^5 + 121 16 X 10 + 88 X 5 + 121 D. 16 X 25 + 35 X 8 + 121 16 X^{25} + 35 X^8 + 121 16 X 25 + 35 X 8 + 121

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Introduction

In mathematics, the process of expanding a squared binomial expression is a fundamental concept that is used extensively in various fields, including algebra, geometry, and calculus. The process of expanding a squared binomial expression involves multiplying the two binomials together using the distributive property. In this article, we will explore the process of expanding the squared binomial expression $\left(4 x^5+11\right)^2$ and determine which of the given expressions is equivalent to it.

Understanding the Concept of Expanding a Squared Binomial Expression

To expand a squared binomial expression, we need to multiply the two binomials together using the distributive property. The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. We can use this property to expand a squared binomial expression by multiplying each term in the first binomial by each term in the second binomial.

Expanding the Squared Binomial Expression

To expand the squared binomial expression $\left(4 x^5+11\right)^2$, we need to multiply the two binomials together using the distributive property. We can start by multiplying the first term in the first binomial, $4 x^5$, by each term in the second binomial, $4 x^5$ and $11$. This gives us:

$$\left(4 x^5\right)\left(4 x^5\right) + \left(4 x^5\right)\left(11\right)$$

Using the distributive property, we can simplify this expression as follows:

$$16 x^{10} + 44 x^5$$

Next, we need to multiply the second term in the first binomial, $11$, by each term in the second binomial, $4 x^5$ and $11$. This gives us:

$$\left(11\right)\left(4 x^5\right) + \left(11\right)\left(11\right)$$

Using the distributive property, we can simplify this expression as follows:

$$44 x^5 + 121$$

Combining the Terms

Now that we have expanded the squared binomial expression, we can combine the terms to get the final result. We can do this by adding the two expressions together:

$$16 x^{10} + 44 x^5 + 44 x^5 + 121$$

Combining like terms, we get:

$$16 x^{10} + 88 x^5 + 121$$

Comparing the Result to the Given Options

Now that we have expanded the squared binomial expression and combined the terms, we can compare the result to the given options to determine which one is equivalent to it. Let's take a look at the options:

A. $16 x^{10} + 121$ B. $16 x^{40} + 121$ C. $16 x^{10} + 88 x^5 + 121$ D. $16 x^{25} + 35 x^8 + 121$

Comparing the result to the given options, we can see that option C is the only one that matches the result.

Conclusion

In this article, we explored the process of expanding a squared binomial expression and determined which of the given expressions is equivalent to $\left(4 x^5+11\right)^2$. We used the distributive property to expand the squared binomial expression and combined the terms to get the final result. We then compared the result to the given options to determine which one is equivalent to it. The correct answer is option C, $16 x^{10} + 88 x^5 + 121$.

Frequently Asked Questions

Q: What is the process of expanding a squared binomial expression? A: The process of expanding a squared binomial expression involves multiplying the two binomials together using the distributive property.

Q: How do I use the distributive property to expand a squared binomial expression? A: To use the distributive property to expand a squared binomial expression, you need to multiply each term in the first binomial by each term in the second binomial.

Q: What is the final result of expanding the squared binomial expression $\left(4 x^5+11\right)^2$? A: The final result of expanding the squared binomial expression $\left(4 x^5+11\right)^2$ is $16 x^{10} + 88 x^5 + 121$.

Key Takeaways

• The process of expanding a squared binomial expression involves multiplying the two binomials together using the distributive property.
• To use the distributive property to expand a squared binomial expression, you need to multiply each term in the first binomial by each term in the second binomial.
• The final result of expanding the squared binomial expression $\left(4 x^5+11\right)^2$ is $16 x^{10} + 88 x^5 + 121$.

Further Reading

If you want to learn more about expanding squared binomial expressions, I recommend checking out the following resources:

• Khan Academy: Expanding Squared Binomials
• Mathway: Expanding Squared Binomials
• Wolfram Alpha: Expanding Squared Binomials

I hope this article has been helpful in understanding the process of expanding a squared binomial expression and determining which of the given expressions is equivalent to $\left(4 x^5+11\right)^2$. If you have any further questions or need additional clarification, please don't hesitate to ask.

Introduction

In our previous article, we explored the process of expanding a squared binomial expression and determined which of the given expressions is equivalent to $\left(4 x^5+11\right)^2$. In this article, we will answer some of the most frequently asked questions about expanding squared binomial expressions.

Q&A

Q: What is the process of expanding a squared binomial expression?

A: The process of expanding a squared binomial expression involves multiplying the two binomials together using the distributive property.

Q: How do I use the distributive property to expand a squared binomial expression?

A: To use the distributive property to expand a squared binomial expression, you need to multiply each term in the first binomial by each term in the second binomial.

Q: What is the difference between expanding a squared binomial expression and expanding a squared trinomial expression?

A: Expanding a squared binomial expression involves multiplying two binomials together, while expanding a squared trinomial expression involves multiplying a trinomial by itself.

Q: Can I use the distributive property to expand a squared binomial expression with variables in the coefficients?

A: Yes, you can use the distributive property to expand a squared binomial expression with variables in the coefficients. For example, if you have the expression $\left(2x^2 + 3\right)^2$, you can expand it using the distributive property.

Q: How do I handle negative coefficients when expanding a squared binomial expression?

A: When expanding a squared binomial expression with negative coefficients, you need to remember that a negative times a negative is a positive. For example, if you have the expression $\left(-2x^2 + 3\right)^2$, you can expand it using the distributive property and remember that the negative coefficients will become positive when multiplied together.

Q: Can I use the distributive property to expand a squared binomial expression with fractional coefficients?

A: Yes, you can use the distributive property to expand a squared binomial expression with fractional coefficients. For example, if you have the expression $\left(\frac{1}{2}x^2 + \frac{1}{3}\right)^2$, you can expand it using the distributive property and remember to multiply the fractions together.

Q: How do I simplify the result of expanding a squared binomial expression?

A: To simplify the result of expanding a squared binomial expression, you need to combine like terms and eliminate any unnecessary parentheses.

Q: Can I use the distributive property to expand a squared binomial expression with complex numbers?

A: Yes, you can use the distributive property to expand a squared binomial expression with complex numbers. For example, if you have the expression $\left(2 + 3i\right)^2$, you can expand it using the distributive property and remember to multiply the complex numbers together.

Conclusion

In this article, we have answered some of the most frequently asked questions about expanding squared binomial expressions. We have covered topics such as the process of expanding a squared binomial expression, using the distributive property, handling negative coefficients, and simplifying the result. We hope that this article has been helpful in understanding the process of expanding squared binomial expressions.

Frequently Asked Questions

Q: What is the process of expanding a squared binomial expression? A: The process of expanding a squared binomial expression involves multiplying the two binomials together using the distributive property.

Q: How do I use the distributive property to expand a squared binomial expression? A: To use the distributive property to expand a squared binomial expression, you need to multiply each term in the first binomial by each term in the second binomial.

Q: What is the difference between expanding a squared binomial expression and expanding a squared trinomial expression? A: Expanding a squared binomial expression involves multiplying two binomials together, while expanding a squared trinomial expression involves multiplying a trinomial by itself.

Key Takeaways

• The process of expanding a squared binomial expression involves multiplying the two binomials together using the distributive property.
• To use the distributive property to expand a squared binomial expression, you need to multiply each term in the first binomial by each term in the second binomial.
• You can use the distributive property to expand a squared binomial expression with variables in the coefficients, negative coefficients, fractional coefficients, and complex numbers.

Further Reading

If you want to learn more about expanding squared binomial expressions, I recommend checking out the following resources:

• Khan Academy: Expanding Squared Binomials
• Mathway: Expanding Squared Binomials
• Wolfram Alpha: Expanding Squared Binomials

I hope this article has been helpful in understanding the process of expanding squared binomial expressions and answering some of the most frequently asked questions. If you have any further questions or need additional clarification, please don't hesitate to ask.