Find The Sum And Classify The Polynomial Based On Degree And Number Of Terms.$3n^2(5n^2 - 2n + 1) + (4n^2 - 11n^4 - 9$\]A. $3^{\text{rd}}$ Degree Polynomial With 4 Terms B. $4^{\text{th}}$ Degree Polynomial With 4 Terms C.

Feb 26, 2025 by ADMIN 225 views

**Solving Polynomial Expressions: Finding the Sum and Classifying the Polynomial**

Understanding Polynomial Expressions

Polynomial expressions are a fundamental concept in algebra, and they play a crucial role in various mathematical operations. A polynomial expression is a mathematical expression consisting of variables, coefficients, and constants, combined using only addition, subtraction, and multiplication. In this article, we will focus on finding the sum of a given polynomial expression and classifying it based on its degree and number of terms.

The Given Polynomial Expression

The given polynomial expression is:

3n^2(5n^2 - 2n + 1) + (4n^2 - 11n^4 - 9)

Distributing and Simplifying the Expression

To find the sum of the given polynomial expression, we need to distribute and simplify it. We will start by distributing the $3n^2$ term to the terms inside the parentheses:

3n^2(5n^2 - 2n + 1) = 15n^4 - 6n^3 + 3n^2

Now, we can rewrite the original expression as:

15n^4 - 6n^3 + 3n^2 + 4n^2 - 11n^4 - 9

Combining Like Terms

Next, we will combine like terms to simplify the expression further. We can combine the $n^4$ terms, $n^3$ terms, and $n^2$ terms separately:

15n^4 - 11n^4 = 4n^4

-6n^3 + 3n^2 + 4n^2 = -6n^3 + 7n^2

Now, we can rewrite the expression as:

4n^4 - 6n^3 + 7n^2 - 9

Classifying the Polynomial

To classify the polynomial, we need to determine its degree and number of terms. The degree of a polynomial is the highest power of the variable (in this case, $n$ ) in the polynomial. The number of terms in a polynomial is the number of individual terms in the polynomial.

Degree of the Polynomial

The highest power of $n$ in the polynomial is $4$ , which is the degree of the polynomial.

Number of Terms

The polynomial has $4$ individual terms: $4n^4$ , $-6n^3$ , $7n^2$ , and $-9$ .

Conclusion

Based on the degree and number of terms, we can classify the polynomial as a $4^{\text{th}}$ degree polynomial with $4$ terms.

Answer

The correct answer is:

B. $4^{\text{th}}$ degree polynomial with $4$ terms

Discussion

Polynomial expressions are a fundamental concept in algebra, and they play a crucial role in various mathematical operations. In this article, we focused on finding the sum of a given polynomial expression and classifying it based on its degree and number of terms. We distributed and simplified the expression, combined like terms, and determined the degree and number of terms of the polynomial. The correct answer is B. $4^{\text{th}}$ degree polynomial with $4$ terms.

Key Takeaways

Polynomial expressions are a fundamental concept in algebra.
To find the sum of a polynomial expression, we need to distribute and simplify it.
To classify a polynomial, we need to determine its degree and number of terms.
The degree of a polynomial is the highest power of the variable in the polynomial.
The number of terms in a polynomial is the number of individual terms in the polynomial.

Real-World Applications

Polynomial expressions have numerous real-world applications in various fields, including:

Physics: Polynomial expressions are used to describe the motion of objects and the behavior of physical systems.
Engineering: Polynomial expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
Computer Science: Polynomial expressions are used in algorithms and data structures to solve complex problems.

Conclusion

In conclusion, polynomial expressions are a fundamental concept in algebra, and they play a crucial role in various mathematical operations. In this article, we focused on finding the sum of a given polynomial expression and classifying it based on its degree and number of terms. We distributed and simplified the expression, combined like terms, and determined the degree and number of terms of the polynomial. The correct answer is B. $4^{\text{th}}$ degree polynomial with $4$ terms.

Q&A: Polynomial Expressions

Q1: What is a polynomial expression?

A polynomial expression is a mathematical expression consisting of variables, coefficients, and constants, combined using only addition, subtraction, and multiplication.

Q2: What is the degree of a polynomial?

The degree of a polynomial is the highest power of the variable (in this case, $n$ ) in the polynomial.

Q3: How do I find the sum of a polynomial expression?

To find the sum of a polynomial expression, you need to distribute and simplify it. This involves multiplying each term inside the parentheses by the term outside the parentheses and then combining like terms.

Q4: What is the difference between a polynomial and a non-polynomial expression?

A polynomial expression is a mathematical expression consisting of variables, coefficients, and constants, combined using only addition, subtraction, and multiplication. A non-polynomial expression, on the other hand, may include other mathematical operations, such as division or exponentiation.

Q5: How do I classify a polynomial based on its degree and number of terms?

To classify a polynomial, you need to determine its degree and number of terms. The degree of a polynomial is the highest power of the variable (in this case, $n$ ) in the polynomial. The number of terms in a polynomial is the number of individual terms in the polynomial.

Q6: What are some real-world applications of polynomial expressions?

Polynomial expressions have numerous real-world applications in various fields, including physics, engineering, and computer science. They are used to describe the motion of objects and the behavior of physical systems, design and optimize systems, and solve complex problems.

Q7: How do I simplify a polynomial expression?

To simplify a polynomial expression, you need to combine like terms. This involves adding or subtracting terms that have the same variable and exponent.

Q8: What is the difference between a linear polynomial and a quadratic polynomial?

A linear polynomial is a polynomial of degree 1, while a quadratic polynomial is a polynomial of degree 2. A linear polynomial has only one term, while a quadratic polynomial has two terms.

Q9: How do I determine the number of terms in a polynomial?

To determine the number of terms in a polynomial, you need to count the individual terms in the polynomial. This involves counting each term separately, including constants.

Q10: What is the importance of polynomial expressions in mathematics?

Polynomial expressions are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. They are used to solve equations, graph functions, and model real-world phenomena.

Conclusion

In conclusion, polynomial expressions are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. In this article, we provided a comprehensive guide to polynomial expressions, including a Q&A section that covers common questions and topics related to polynomial expressions.

Key Takeaways

Polynomial expressions are a fundamental concept in mathematics.
To find the sum of a polynomial expression, you need to distribute and simplify it.
To classify a polynomial, you need to determine its degree and number of terms.
Polynomial expressions have numerous real-world applications in various fields.
To simplify a polynomial expression, you need to combine like terms.

Real-World Applications

Polynomial expressions have numerous real-world applications in various fields, including:

Physics: Polynomial expressions are used to describe the motion of objects and the behavior of physical systems.
Engineering: Polynomial expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
Computer Science: Polynomial expressions are used in algorithms and data structures to solve complex problems.