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. $25 Z^4+30 Z^3+29 Z^2+12 Z+4$

$$

Understanding the Problem

To find the equivalent expression for $\left(5 z^2+3 z+2\right)^2$, we need to expand the given expression using the formula for squaring a binomial. This involves multiplying the binomial by itself and then simplifying the resulting expression.

Expanding the Expression

The given expression is $\left(5 z^2+3 z+2\right)^2$. To expand this, we can use the formula for squaring a binomial:

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

In this case, $a = 5z^2$ and $b = 3z + 2$. Substituting these values into the formula, we get:

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

Simplifying the Expression

Now, we can simplify each term in the expression:

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

${(3z + 2)^2 = 9z^2 + 12z + 4}$

The middle term requires us to use the distributive property to multiply $5z^2$ by $3z$ and $5z^2$ by $2$:

${2(5z^2)(3z + 2) = 2(15z^3 + 10z^2) = 30z^3 + 20z^2}$

Combining the Terms

Now, we can combine the simplified terms to get the final expression:

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

Combining like terms, we get:

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

Conclusion

The equivalent expression for $\left(5 z^2+3 z+2\right)^2$ is $25 z^4+30 z^3+29 z^2+12 z+4$. This can be verified by expanding the given expression using the formula for squaring a binomial and simplifying the resulting expression.

Answer

The correct answer is D. $25 z^4+30 z^3+29 z^2+12 z+4$.

Final Thoughts

In this problem, we used the formula for squaring a binomial to expand the given expression and then simplified the resulting expression to find the equivalent expression. This problem requires a good understanding of algebraic expressions and the ability to apply formulas to solve problems.

$$

Understanding the Problem

To find the equivalent expression for $\left(5 z^2+3 z+2\right)^2$, we need to expand the given expression using the formula for squaring a binomial. This involves multiplying the binomial by itself and then simplifying the resulting expression.

Q&A

Q: What is the formula for squaring a binomial?

A: The formula for squaring a binomial is:

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

Q: How do we apply the formula to the given expression?

A: To apply the formula, we need to identify the values of $a$ and $b$ in the given expression. In this case, $a = 5z^2$ and $b = 3z + 2$. We can then substitute these values into the formula and simplify the resulting expression.

Q: What is the first term in the expanded expression?

A: The first term in the expanded expression is $(5z^2)^2 = 25z^4$.

Q: What is the second term in the expanded expression?

A: The second term in the expanded expression is $2(5z^2)(3z + 2) = 30z^3 + 20z^2$.

Q: What is the third term in the expanded expression?

A: The third term in the expanded expression is $(3z + 2)^2 = 9z^2 + 12z + 4$.

Q: How do we combine the terms in the expanded expression?

A: To combine the terms, we need to add the like terms together. In this case, we have:

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

Combining like terms, we get:

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

Q: What is the final answer?

A: The final answer is $25 z^4+30 z^3+29 z^2+12 z+4$.

Conclusion

In this article, we used the formula for squaring a binomial to expand the given expression and then simplified the resulting expression to find the equivalent expression. We also answered some common questions related to the problem to help readers understand the solution better.

Final Thoughts

In this problem, we used the formula for squaring a binomial to expand the given expression and then simplified the resulting expression to find the equivalent expression. This problem requires a good understanding of algebraic expressions and the ability to apply formulas to solve problems.

Related Questions

• What is the formula for squaring a binomial?
• How do we apply the formula to the given expression?
• What is the first term in the expanded expression?
• What is the second term in the expanded expression?
• What is the third term in the expanded expression?
• How do we combine the terms in the expanded expression?
• What is the final answer?

Related Articles

• Understanding Algebraic Expressions
• Applying Formulas to Solve Problems
• Simplifying Algebraic Expressions