Multiplying the Result by the Third Polynomial

Oct 13, 2025 by windailys.com 129 views

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Understanding the Problem

When it comes to multiplying polynomials, it's essential to understand the concept of the distributive property. This property allows us to multiply each term in one polynomial by each term in another polynomial. In this article, we'll explore how to multiply polynomials and find the equivalent expression.

The Distributive Property

The distributive property states that for any real numbers a, b, and c:

a(b + c) = ab + ac

This property can be extended to polynomials, where each term in one polynomial is multiplied by each term in another polynomial.

Multiplying Polynomials

To multiply polynomials, we'll use the distributive property to multiply each term in one polynomial by each term in another polynomial. Let's consider the given expression:

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

We'll start by multiplying the first two polynomials:

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

Using the distributive property, we get:

$p^3 \cdot 2 p^2 = 2 p^5$

$p^3 \cdot -4 p = -4 p^4$

So, the result of multiplying the first two polynomials is:

$2 p^5 - 4 p^4$

Multiplying the Result by the Third Polynomial

Now, we'll multiply the result by the third polynomial:

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

Using the distributive property again, we get:

$\left(2 p^5\right)\left(3 p^2-1\right) = 6 p^7 - 2 p^5$

$\left(-4 p^4\right)\left(3 p^2-1\right) = -12 p^6 + 4 p^4$

So, the final result is:

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

Comparing the Results

Now, let's compare the result we obtained with the given options:

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

Our result is:

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

Comparing the results, we can see that option A is the correct answer.

Conclusion

Multiplying polynomials can be a challenging task, but by using the distributive property, we can simplify the process. In this article, we explored how to multiply polynomials and found the equivalent expression. We also compared the result with the given options and determined that option A is the correct answer.

Tips and Tricks

When multiplying polynomials, use the distributive property to multiply each term in one polynomial by each term in another polynomial.
Start by multiplying the first two polynomials and then multiply the result by the third polynomial.
Use the distributive property again to multiply each term in the result by each term in the third polynomial.
Simplify the result by combining like terms.

By following these tips and tricks, you can become more confident in your ability to multiply polynomials and find the equivalent expression.

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Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows us to multiply each term in one polynomial by each term in another polynomial. It states that for any real numbers a, b, and c:

a(b + c) = ab + ac

This property can be extended to polynomials, where each term in one polynomial is multiplied by each term in another polynomial.

Q: How do I multiply polynomials?

A: To multiply polynomials, you'll use the distributive property to multiply each term in one polynomial by each term in another polynomial. Here's a step-by-step guide:

Multiply the first two polynomials using the distributive property.
Multiply the result by the third polynomial using the distributive property again.
Simplify the result by combining like terms.

Q: What is the difference between multiplying polynomials and multiplying numbers?

A: Multiplying polynomials is similar to multiplying numbers, but with polynomials, you'll be multiplying each term in one polynomial by each term in another polynomial. This can result in a more complex expression with multiple terms.

Q: How do I simplify the result of multiplying polynomials?

A: To simplify the result of multiplying polynomials, you'll need to combine like terms. This involves adding or subtracting terms that have the same variable and exponent.

For example, if you have the expression:

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

You can combine the like terms by adding or subtracting the coefficients:

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

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

$= 2 p^4$

Q: What are some common mistakes to avoid when multiplying polynomials?

A: Here are some common mistakes to avoid when multiplying polynomials:

Forgetting to use the distributive property when multiplying polynomials.
Not combining like terms when simplifying the result.
Making errors when multiplying or adding terms.
Not checking the result for errors or inconsistencies.

Q: How can I practice multiplying polynomials?

A: There are many ways to practice multiplying polynomials, including:

Using online resources or worksheets to practice multiplying polynomials.
Working with a tutor or teacher to get one-on-one practice.
Joining a study group or math club to practice with others.
Using real-world examples or applications to practice multiplying polynomials.

Q: What are some real-world applications of multiplying polynomials?

A: Multiplying polynomials has many real-world applications, including:

Calculating the area or volume of complex shapes.
Modeling population growth or decline.
Analyzing data or trends in business or economics.
Solving problems in physics or engineering.

By practicing multiplying polynomials and understanding its real-world applications, you can become more confident in your ability to solve complex math problems.