A Rectangle Has A Height Of $n^3 + 4n^2 + 3n$ And A Width Of $n^3 + 5n^2$.Express The Area Of The Entire Rectangle. Your Answer Should Be A Polynomial In Standard Form.Area $= \square$

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Introduction

In mathematics, the area of a rectangle is calculated by multiplying its height and width. In this problem, we are given the height and width of a rectangle as polynomials in terms of $n$. Our goal is to express the area of the entire rectangle as a polynomial in standard form.

Calculating the Area

To calculate the area of the rectangle, we need to multiply its height and width. The height of the rectangle is given by the polynomial $n^3 + 4n^2 + 3n$, and the width is given by the polynomial $n^3 + 5n^2$. We can multiply these two polynomials using the distributive property.

Multiplying Polynomials

When multiplying two polynomials, we need to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms. In this case, we have:

$(n^3 + 4n^2 + 3n) \times (n^3 + 5n^2)$

Using the distributive property, we can expand this product as follows:

$n^3 \times n^3 + n^3 \times 5n^2 + n^3 \times 3n + 4n^2 \times n^3 + 4n^2 \times 5n^2 + 4n^2 \times 3n + 3n \times n^3 + 3n \times 5n^2 + 3n \times 3n$

Combining Like Terms

Now, we can combine like terms in the expanded product:

$n^6 + 5n^5 + 3n^4 + 4n^5 + 20n^4 + 12n^3 + 3n^4 + 15n^3 + 9n^2$

Combining like terms, we get:

$n^6 + (5n^5 + 4n^5) + (3n^4 + 20n^4 + 3n^4) + (12n^3 + 15n^3) + 9n^2$

Simplifying further, we get:

$n^6 + 9n^5 + 26n^4 + 27n^3 + 9n^2$

Conclusion

In this problem, we were given the height and width of a rectangle as polynomials in terms of $n$. We calculated the area of the rectangle by multiplying its height and width and then combining like terms. The area of the rectangle is given by the polynomial $n^6 + 9n^5 + 26n^4 + 27n^3 + 9n^2$.

Final Answer

The area of the rectangle is $n^6 + 9n^5 + 26n^4 + 27n^3 + 9n^2$.

Discussion

This problem is a good example of how to multiply polynomials and combine like terms. It also shows how to calculate the area of a rectangle when the height and width are given as polynomials. In real-world applications, this type of problem can arise in engineering, physics, and other fields where polynomials are used to model real-world phenomena.

Related Problems

• Calculating the area of a triangle with vertices at $(0,0)$, $(a,0)$, and $(0,b)$.
• Finding the volume of a solid with a base area given by a polynomial and a height given by a polynomial.
• Solving a system of polynomial equations.

References

• Polynomial Multiplication
• Area of a Rectangle
• Polynomial Equations

Tags

• mathematics
• polynomials
• area of a rectangle
• polynomial multiplication
• like terms
• algebra
• geometry
  

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Q&A

Q: What is the formula for calculating the area of a rectangle?

A: The formula for calculating the area of a rectangle is given by the product of its height and width.

Q: How do I multiply polynomials?

A: To multiply polynomials, you need to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable raised to the same power, while unlike terms are terms that have different variables or different powers of the same variable.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you need to combine like terms and eliminate any unnecessary terms.

Q: What is the final answer to the problem?

A: The final answer to the problem is $n^6 + 9n^5 + 26n^4 + 27n^3 + 9n^2$.

Q: Can you explain the concept of polynomial multiplication in more detail?

A: Polynomial multiplication is a process of multiplying two or more polynomials together. It involves multiplying each term in the first polynomial by each term in the second polynomial and then combining like terms.

Q: How do I apply polynomial multiplication in real-world problems?

A: Polynomial multiplication is used in a variety of real-world problems, including engineering, physics, and computer science. It is used to model and analyze complex systems, and to solve problems that involve multiple variables.

Q: What are some common applications of polynomial multiplication?

A: Some common applications of polynomial multiplication include:

• Calculating the area of a rectangle with a height and width given by polynomials
• Finding the volume of a solid with a base area given by a polynomial and a height given by a polynomial
• Solving a system of polynomial equations
• Modeling and analyzing complex systems in engineering, physics, and computer science

Q: Can you provide some examples of polynomial multiplication?

A: Here are a few examples of polynomial multiplication:

• $(x^2 + 3x + 2) \times (x + 1)$
• $(x^3 + 2x^2 + x) \times (x^2 + 3x + 2)$
• $(x^4 + 3x^3 + 2x^2) \times (x + 1)$

Q: How do I use polynomial multiplication to solve a system of polynomial equations?

A: To use polynomial multiplication to solve a system of polynomial equations, you need to multiply each equation by a common factor and then combine like terms. This will allow you to eliminate variables and solve for the remaining variables.

Q: What are some common mistakes to avoid when using polynomial multiplication?

A: Some common mistakes to avoid when using polynomial multiplication include:

• Failing to combine like terms
• Failing to eliminate unnecessary terms
• Failing to check for errors in the multiplication process

Q: Can you provide some tips for using polynomial multiplication effectively?

A: Here are a few tips for using polynomial multiplication effectively:

• Make sure to combine like terms carefully
• Eliminate unnecessary terms to simplify the expression
• Check for errors in the multiplication process
• Use polynomial multiplication to solve a system of polynomial equations

Conclusion

In this article, we have discussed the concept of polynomial multiplication and how it is used to calculate the area of a rectangle with a height and width given by polynomials. We have also provided some examples of polynomial multiplication and discussed some common applications of the concept. Additionally, we have provided some tips for using polynomial multiplication effectively and some common mistakes to avoid.