Find The Difference Of The Polynomials Given Below And Classify It In Terms Of Degree And Number Of Terms.${ 3n 2(n 2+4n-5) - (2n 2-n 4+3) }$A. ${ 3^{\text{rd}}\$} Degree Polynomial With 4 Terms B. ${ 4^{\text{th}}\$}

by ADMIN 220 views

Introduction

Polynomials are a fundamental concept in algebra, and understanding how to simplify and classify them is crucial for solving various mathematical problems. In this article, we will focus on finding the difference of two given polynomials and classifying it in terms of degree and number of terms.

The Given Polynomials

The two polynomials given are:

3n2(n2+4n−5)−(2n2−n4+3){ 3n^2(n^2+4n-5) - (2n^2-n^4+3) }

To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses.
  2. Multiply the terms.
  3. Combine like terms.

Simplifying the Expression

Let's start by simplifying the expression inside the first set of parentheses:

n2+4n−5{ n^2+4n-5 }

This expression cannot be simplified further.

Next, let's simplify the expression inside the second set of parentheses:

2n2−n4+3{ 2n^2-n^4+3 }

This expression can be simplified by combining like terms:

−n4+2n2+3{ -n^4+2n^2+3 }

Now, let's rewrite the original expression using the simplified expressions:

3n2(n2+4n−5)−(−n4+2n2+3){ 3n^2(n^2+4n-5) - (-n^4+2n^2+3) }

Distributing and Combining Like Terms

To simplify the expression further, we need to distribute the terms and combine like terms:

3n2(n2+4n−5)=3n4+12n3−15n2{ 3n^2(n^2+4n-5) = 3n^4+12n^3-15n^2 }

−(−n4+2n2+3)=n4−2n2−3{ -(-n^4+2n^2+3) = n^4-2n^2-3 }

Now, let's combine the two simplified expressions:

3n4+12n3−15n2+n4−2n2−3{ 3n^4+12n^3-15n^2+n^4-2n^2-3 }

Combine like terms:

4n4+12n3−17n2−3{ 4n^4+12n^3-17n^2-3 }

Classifying the Polynomial

Now that we have simplified the expression, let's classify it in terms of degree and number of terms.

Degree of the Polynomial

The degree of a polynomial is the highest power of the variable (in this case, n). In the simplified expression, the highest power of n is 4. Therefore, the degree of the polynomial is 4.

Number of Terms

The number of terms in a polynomial is the number of individual terms. In the simplified expression, there are 4 individual terms:

  1. 4n^4
  2. 12n^3
  3. -17n^2
  4. -3

Therefore, the polynomial has 4 terms.

Conclusion

In conclusion, the difference of the two given polynomials is a 4th degree polynomial with 4 terms. This classification is crucial for solving various mathematical problems, and understanding how to simplify and classify polynomials is essential for success in algebra and beyond.

Final Answer

The final answer is:

A. 4th degree polynomial with 4 terms

Discussion

This problem requires a deep understanding of algebraic expressions and the ability to simplify and classify polynomials. The key concepts involved in this problem include:

  • Distributing and combining like terms
  • Evaluating expressions inside parentheses
  • Identifying the degree and number of terms in a polynomial

Introduction

In our previous article, we explored the process of simplifying and classifying polynomials. We walked through a step-by-step guide on how to simplify the difference of two given polynomials and classify it in terms of degree and number of terms. In this article, we will address some common questions and concerns related to polynomial simplification and classification.

Q&A

Q: What is the difference between a polynomial and an expression?

A: A polynomial is a mathematical expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. An expression, on the other hand, can include any combination of variables, coefficients, and mathematical operations.

Q: How do I determine the degree of a polynomial?

A: To determine the degree of a polynomial, you need to identify the highest power of the variable (in this case, n). In the simplified expression, the highest power of n is 4. Therefore, the degree of the polynomial is 4.

Q: What is the significance of the degree of a polynomial?

A: The degree of a polynomial is crucial in determining its behavior and properties. For example, a polynomial of degree 1 is a linear function, while a polynomial of degree 2 is a quadratic function. The degree of a polynomial also affects its graph and its ability to intersect with other curves.

Q: How do I classify a polynomial in terms of its number of terms?

A: To classify a polynomial in terms of its number of terms, you need to count the individual terms in the polynomial. In the simplified expression, there are 4 individual terms:

  1. 4n^4
  2. 12n^3
  3. -17n^2
  4. -3

Therefore, the polynomial has 4 terms.

Q: What is the difference between a monomial, binomial, and polynomial?

A: A monomial is a single term that consists of a variable and a coefficient. A binomial is a polynomial with two terms. A polynomial is a mathematical expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: How do I simplify a polynomial with multiple variables?

A: To simplify a polynomial with multiple variables, you need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside parentheses.
  2. Multiply the terms.
  3. Combine like terms.

For example, consider the polynomial:

2x2y+3x2y−4xy{ 2x^2y + 3x^2y - 4xy }

To simplify this polynomial, you need to combine like terms:

2x2y+3x2y−4xy=5x2y−4xy{ 2x^2y + 3x^2y - 4xy = 5x^2y - 4xy }

Q: What is the significance of the leading coefficient in a polynomial?

A: The leading coefficient is the coefficient of the highest power of the variable in a polynomial. The leading coefficient affects the behavior and properties of the polynomial. For example, a polynomial with a positive leading coefficient will have a positive leading term, while a polynomial with a negative leading coefficient will have a negative leading term.

Q: How do I determine the number of solutions to a polynomial equation?

A: To determine the number of solutions to a polynomial equation, you need to examine the degree of the polynomial. A polynomial of degree 1 has one solution, while a polynomial of degree 2 has two solutions. A polynomial of degree 3 has three solutions, and so on.

Conclusion

In conclusion, polynomial simplification and classification are essential skills for success in algebra and beyond. By understanding the concepts and techniques outlined in this article, you can develop a strong foundation in algebra and improve your problem-solving skills.