Which Of The Following Is Equivalent To $\left(p^3\right)\left(2 P^2-4 P\right)\left(3 P^2-1\right$\]?A. $\left(p^3\right)\left(6 P^4-12 P^3-2 P^2+4 P\right$\]B. $\left(p^3\right)\left(6 P^4+4 P\right$\]C. $\left(2 P^6-4

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Introduction

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In algebra, multiplying polynomials is a fundamental concept that helps us simplify complex expressions and solve equations. When we multiply polynomials, we need to follow a specific set of rules to ensure that we get the correct result. In this article, we will explore the process of multiplying polynomials and apply it to a given problem.

What are Polynomials?

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A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be written in the form of:

$$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$$

where $a_n$, $a_{n-1}$, $\ldots$, $a_1$, and $a_0$ are constants, and $x$ is the variable.

Multiplying Polynomials

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To multiply polynomials, we need to follow the distributive property, which states that:

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

Using this property, we can multiply each term in the first polynomial by each term in the second polynomial.

Example Problem

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Let's consider the following problem:

$$\left(p^3\right)\left(2 p^2-4 p\right)\left(3 p^2-1\right)$$

We need to multiply these three polynomials together to get the final result.

Step 1: Multiply the First Two Polynomials

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To multiply the first two polynomials, we need to multiply each term in the first polynomial by each term in the second polynomial.

$$\left(p^3\right)\left(2 p^2-4 p\right) = p^3(2 p^2) + p^3(-4 p)$$

$$= 2 p^5 - 4 p^4$$

Step 2: Multiply the Result by the Third Polynomial

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Now, we need to multiply the result from Step 1 by the third polynomial.

$$\left(2 p^5 - 4 p^4\right)\left(3 p^2-1\right) = \left(2 p^5\right)\left(3 p^2-1\right) + \left(-4 p^4\right)\left(3 p^2-1\right)$$

$$= 6 p^7 - 2 p^5 - 12 p^6 + 4 p^4$$

Simplifying the Result

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Now, we can simplify the result by combining like terms.

$$6 p^7 - 2 p^5 - 12 p^6 + 4 p^4 = 6 p^7 - 12 p^6 - 2 p^5 + 4 p^4$$

Conclusion

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In conclusion, multiplying polynomials requires us to follow a specific set of rules to ensure that we get the correct result. By applying the distributive property and combining like terms, we can simplify complex expressions and solve equations.

Answer

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Based on the steps outlined above, the final result is:

$$6 p^7 - 12 p^6 - 2 p^5 + 4 p^4$$

This is equivalent to option A.

Comparison with Other Options

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Let's compare our result with the other options:

• Option B: $\left(p^3\right)\left(6 p^4+4 p\right)$
• Option C: $\left(2 p^6-4 p^4\right)$

Our result is different from both options B and C.

Final Answer

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Based on the steps outlined above, the final answer is:

• A. $\left(p^3\right)\left(6 p^4-12 p^3-2 p^2+4 p\right)$

This is the correct answer.

Key Takeaways

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• Multiplying polynomials requires us to follow a specific set of rules to ensure that we get the correct result.
• We need to apply the distributive property and combine like terms to simplify complex expressions.
• The final result is equivalent to option A.

Practice Problems

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Try the following practice problems to test your understanding:

• Multiply the polynomials: $\left(p^2\right)\left(2 p-3\right)\left(p+2\right)$
• Multiply the polynomials: $\left(2 p^3-3 p^2\right)\left(p^2+2 p-1\right)$

Conclusion

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In conclusion, multiplying polynomials is a fundamental concept in algebra that requires us to follow a specific set of rules to ensure that we get the correct result. By applying the distributive property and combining like terms, we can simplify complex expressions and solve equations.

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Q&A: Multiplying Polynomials

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Q: What is the distributive property?

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A: The distributive property is a mathematical concept that states that:

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

This property allows us to multiply each term in the first polynomial by each term in the second polynomial.

Q: How do I multiply polynomials?

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A: To multiply polynomials, you need to follow these steps:

1. Multiply each term in the first polynomial by each term in the second polynomial.
2. Combine like terms to simplify the result.

Q: What are like terms?

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A: Like terms are terms that have the same variable and exponent. For example, $2p^2$ and $3p^2$ are like terms because they both have the variable $p$ and the exponent $2$.

Q: How do I simplify the result?

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A: To simplify the result, you need to combine like terms. This means adding or subtracting the coefficients of like terms.

Q: What is the final result of multiplying the polynomials $\left(p^3\right)\left(2 p^2-4 p\right)\left(3 p^2-1\right)$?

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A: The final result is:

$$6 p^7 - 12 p^6 - 2 p^5 + 4 p^4$$

Q: Is the final result equivalent to option A?

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A: Yes, the final result is equivalent to option A.

Q: What is the difference between option B and the final result?

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A: Option B is:

$$\left(p^3\right)\left(6 p^4+4 p\right)$$

This is different from the final result because it has a different coefficient for the $p^4$ term.

Q: What is the difference between option C and the final result?

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A: Option C is:

$$\left(2 p^6-4 p^4\right)$$

This is different from the final result because it has a different coefficient for the $p^6$ term.

Q: Can I use the distributive property to multiply polynomials with more than two terms?

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A: Yes, you can use the distributive property to multiply polynomials with more than two terms. Simply multiply each term in the first polynomial by each term in the second polynomial, and then combine like terms.

Q: How do I multiply polynomials with negative coefficients?

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A: To multiply polynomials with negative coefficients, you need to follow the same steps as multiplying polynomials with positive coefficients. The only difference is that you will have negative coefficients in the result.

Q: Can I use a calculator to multiply polynomials?

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A: Yes, you can use a calculator to multiply polynomials. However, it's always a good idea to double-check your work by multiplying the polynomials by hand.

Conclusion

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In conclusion, multiplying polynomials is a fundamental concept in algebra that requires us to follow a specific set of rules to ensure that we get the correct result. By applying the distributive property and combining like terms, we can simplify complex expressions and solve equations.

Practice Problems

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Try the following practice problems to test your understanding:

• Multiply the polynomials: $\left(p^2\right)\left(2 p-3\right)\left(p+2\right)$
• Multiply the polynomials: $\left(2 p^3-3 p^2\right)\left(p^2+2 p-1\right)$

Key Takeaways

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• Multiplying polynomials requires us to follow a specific set of rules to ensure that we get the correct result.
• We need to apply the distributive property and combine like terms to simplify complex expressions.
• The final result is equivalent to option A.

Final Answer

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Based on the steps outlined above, the final answer is:

• A. $\left(p^3\right)\left(6 p^4-12 p^3-2 p^2+4 p\right)$

This is the correct answer.