What Expression Is Equivalent To $\left(5 Z^2+3 Z+2\right)^2$?A. $5 Z^4+3 Z^2+4$B. $5 Z^4+9 Z^2+4$C. $25 Z^4+30 Z^3+19 Z^2+12 Z+4$D. $25 Z^4+30 Z^3+29 Z^2+12 Z+4$

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Introduction

In algebra, the process of expanding a squared binomial is a crucial concept that helps us simplify complex expressions. When we square a binomial, we need to multiply each term in the binomial by every other term, including itself. This process can be time-consuming and error-prone, especially when dealing with polynomials of higher degrees. In this article, we will explore the process of expanding a squared binomial and apply it to the given expression $\left(5 z^2+3 z+2\right)^2$. We will also examine the answer choices and determine which one is equivalent to the given expression.

The Process of Expanding a Squared Binomial

To expand a squared binomial, we need to follow the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. When we square a binomial, we can use this property to multiply each term in the binomial by every other term.

Let's consider the binomial $a + b$. When we square it, we get:

$(a + b)^2 = a^2 + 2ab + b^2$

Notice that we have three terms: $a^2$, $2ab$, and $b^2$. These terms are obtained by multiplying each term in the binomial by every other term.

Applying the Process to the Given Expression

Now, let's apply the process of expanding a squared binomial to the given expression $\left(5 z^2+3 z+2\right)^2$. We can use the distributive property to multiply each term in the binomial by every other term.

$(5 z^2+3 z+2)^2 = (5 z^2)^2 + 2(5 z^2)(3 z) + 2(5 z^2)(2) + (3 z)^2 + 2(3 z)(2) + (2)^2$

Expanding each term, we get:

$(5 z^2)^2 = 25 z^4$

$2(5 z^2)(3 z) = 30 z^3$

$2(5 z^2)(2) = 20 z^2$

$(3 z)^2 = 9 z^2$

$2(3 z)(2) = 12 z$

$(2)^2 = 4$

Now, let's combine like terms:

$25 z^4 + 30 z^3 + 20 z^2 + 9 z^2 + 12 z + 4$

Combining like terms, we get:

$25 z^4 + 30 z^3 + 29 z^2 + 12 z + 4$

Comparing the Result with the Answer Choices

Now, let's compare the result we obtained with the answer choices:

A. $5 z^4+3 z^2+4$

B. $5 z^4+9 z^2+4$

C. $25 z^4+30 z^3+19 z^2+12 z+4$

D. $25 z^4+30 z^3+29 z^2+12 z+4$

The only answer choice that matches our result is:

D. $25 z^4+30 z^3+29 z^2+12 z+4$

Therefore, the expression equivalent to $\left(5 z^2+3 z+2\right)^2$ is:

$25 z^4+30 z^3+29 z^2+12 z+4$

Conclusion

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Introduction

In our previous article, we explored the process of expanding a squared binomial and applied it to the expression $\left(5 z^2+3 z+2\right)^2$. We used the distributive property to multiply each term in the binomial by every other term and obtained the result $25 z^4+30 z^3+29 z^2+12 z+4$. In this article, we will answer some frequently asked questions about expanding squared binomials.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. This property allows us to multiply each term in a binomial by every other term.

Q: How do I expand a squared binomial?

A: To expand a squared binomial, you need to follow these steps:

1. Multiply each term in the binomial by every other term.
2. Use the distributive property to simplify the expression.
3. Combine like terms to obtain the final result.

Q: What is the difference between expanding a squared binomial and multiplying two binomials?

A: Expanding a squared binomial involves multiplying each term in the binomial by every other term, whereas multiplying two binomials involves multiplying each term in one binomial by every term in the other binomial.

Q: Can I use the FOIL method to expand a squared binomial?

A: Yes, you can use the FOIL method to expand a squared binomial. The FOIL method involves multiplying the first terms in each binomial, then multiplying the outer terms, then multiplying the inner terms, and finally multiplying the last terms.

Q: What is the FOIL method?

A: The FOIL method is a mnemonic device that helps you remember the steps involved in multiplying two binomials. FOIL stands for First, Outer, Inner, Last, and it reminds you to multiply the first terms, then the outer terms, then the inner terms, and finally the last terms.

Q: Can I use the distributive property to expand a squared binomial with more than two terms?

A: Yes, you can use the distributive property to expand a squared binomial with more than two terms. Simply multiply each term in the binomial by every other term, and then combine like terms to obtain the final result.

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

A: Expanding a squared binomial involves multiplying each term in the binomial by every other term, whereas expanding a cubed binomial involves multiplying each term in the binomial by every other term, and then multiplying the result by the third term.

Q: Can I use the distributive property to expand a cubed binomial?

A: Yes, you can use the distributive property to expand a cubed binomial. Simply multiply each term in the binomial by every other term, and then multiply the result by the third term.

Conclusion

In this article, we answered some frequently asked questions about expanding squared binomials. We discussed the distributive property, the FOIL method, and the process of expanding a squared binomial with more than two terms. We also compared expanding a squared binomial with expanding a cubed binomial. We hope this article has been helpful in clarifying any confusion you may have had about expanding squared binomials.

Additional Resources

If you are looking for additional resources on expanding squared binomials, we recommend the following:

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

These resources provide step-by-step instructions and examples on how to expand squared binomials.