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. { 4^{\text{th}} $}$ Degree Polynomial With 4 Terms B. [$ 3^{\text{rd}}

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Introduction

In algebra, polynomials are mathematical expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When we are given two polynomials, we can find their difference by subtracting one from the other. In this article, we will find the difference of the given polynomials and classify it in terms of degree and number of terms.

Given Polynomials

The given polynomials are:

3n2(n2+4nβˆ’5)βˆ’(2n2βˆ’n4+3){ 3n^2(n^2 + 4n - 5) - (2n^2 - n^4 + 3) }

To find the difference, we need to expand and simplify the expression.

Step 1: Expand the Expression

First, let's expand the expression using the distributive property.

3n2(n2+4nβˆ’5)βˆ’(2n2βˆ’n4+3){ 3n^2(n^2 + 4n - 5) - (2n^2 - n^4 + 3) }

=3n2(n2)+3n2(4n)+3n2(βˆ’5)βˆ’2n2+n4βˆ’3{ = 3n^2(n^2) + 3n^2(4n) + 3n^2(-5) - 2n^2 + n^4 - 3 }

=3n4+12n3βˆ’15n2βˆ’2n2+n4βˆ’3{ = 3n^4 + 12n^3 - 15n^2 - 2n^2 + n^4 - 3 }

Step 2: Combine Like Terms

Now, let's combine like terms.

=3n4+12n3βˆ’15n2βˆ’2n2+n4βˆ’3{ = 3n^4 + 12n^3 - 15n^2 - 2n^2 + n^4 - 3 }

=(3n4+n4)+(12n3)+(βˆ’15n2βˆ’2n2)βˆ’3{ = (3n^4 + n^4) + (12n^3) + (-15n^2 - 2n^2) - 3 }

=4n4+12n3βˆ’17n2βˆ’3{ = 4n^4 + 12n^3 - 17n^2 - 3 }

Step 3: Classify the Polynomial

Now that we have the expanded and simplified 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 any term. In the expression 4n^4 + 12n^3 - 17n^2 - 3, 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 expression. In the expression 4n^4 + 12n^3 - 17n^2 - 3, there are 4 individual terms. Therefore, the number of terms in the polynomial is 4.

Conclusion

In conclusion, the difference of the given polynomials is 4n^4 + 12n^3 - 17n^2 - 3. The degree of the polynomial is 4, and the number of terms is 4.

Answer

The correct answer is:

A. 4th degree polynomial with 4 terms

Discussion

This problem requires the student to find the difference of two polynomials and classify it in terms of degree and number of terms. The student needs to expand and simplify the expression using the distributive property and combine like terms. The student also needs to identify the degree and number of terms in the resulting polynomial.

Tips and Tricks

  • When finding the difference of two polynomials, make sure to expand and simplify the expression using the distributive property.
  • When combining like terms, make sure to group terms with the same variable and exponent.
  • When classifying the polynomial, make sure to identify the degree and number of terms in the resulting polynomial.

Practice Problems

  1. Find the difference of the polynomials 2x^2 + 3x - 1 and x^2 - 2x + 4.
  2. Classify the polynomial 3x^3 + 2x^2 - 5x + 1 in terms of degree and number of terms.
  3. Find the difference of the polynomials x^2 + 2x - 3 and 2x^2 - 3x + 1.

References

  • [Algebra textbook]
  • [Online algebra resources]

Introduction

In our previous article, we discussed how to find the difference of two polynomials and classify it in terms of degree and number of terms. In this article, we will provide a Q&A section to help you better understand the concept and provide additional practice problems.

Q: What is the difference of two polynomials?

A: The difference of two polynomials is the result of subtracting one polynomial from another. It is obtained by subtracting each term of the second polynomial from the corresponding term of the first polynomial.

Q: How do I find the difference of two polynomials?

A: To find the difference of two polynomials, you need to follow these steps:

  1. Expand and simplify the expression using the distributive property.
  2. Combine like terms.
  3. Identify the degree and number of terms in the resulting polynomial.

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest power of the variable (in this case, n) in any term. It is denoted by the exponent of the highest power of the variable.

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

A: To classify a polynomial in terms of degree and number of terms, you need to follow these steps:

  1. Identify the highest power of the variable in any term (degree).
  2. Count the number of individual terms in the polynomial (number of terms).

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

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A rational expression is an expression consisting of a fraction of two polynomials.

Q: Can I use the distributive property to simplify a rational expression?

A: No, the distributive property can only be used to simplify polynomials, not rational expressions.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

  1. Factor the numerator and denominator.
  2. Cancel out any common factors.
  3. Simplify the resulting expression.

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

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. An algebraic expression is a more general term that includes polynomials, rational expressions, and other types of expressions.

Practice Problems

  1. Find the difference of the polynomials 2x^2 + 3x - 1 and x^2 - 2x + 4.
  2. Classify the polynomial 3x^3 + 2x^2 - 5x + 1 in terms of degree and number of terms.
  3. Find the difference of the polynomials x^2 + 2x - 3 and 2x^2 - 3x + 1.
  4. Simplify the rational expression (x^2 + 2x - 3) / (x + 1).
  5. Classify the algebraic expression 2x^2 + 3x - 1 in terms of degree and number of terms.

Answers

  1. 3x^2 + 5x - 5
  2. 4th degree polynomial with 4 terms
  3. 3x^2 - x - 1
  4. x - 2
  5. 2nd degree polynomial with 3 terms

Discussion

This Q&A section provides additional practice problems and answers to help you better understand the concept of finding the difference of polynomials and classifying in terms of degree and number of terms. The practice problems cover a range of topics, including simplifying rational expressions and classifying algebraic expressions.

Tips and Tricks

  • When finding the difference of two polynomials, make sure to expand and simplify the expression using the distributive property.
  • When combining like terms, make sure to group terms with the same variable and exponent.
  • When classifying a polynomial, make sure to identify the degree and number of terms in the resulting polynomial.
  • When simplifying a rational expression, make sure to factor the numerator and denominator and cancel out any common factors.

References

  • [Algebra textbook]
  • [Online algebra resources]

Note: The references provided are fictional and for demonstration purposes only.