Determining the Nature of the Sum

Oct 13, 2025 by magicalceylontours.com 49 views

Introduction

When dealing with polynomials, understanding their properties and behavior is crucial in various mathematical applications. In this context, we are given two polynomials and asked to determine the nature of their sum. The two polynomials are:

{ \begin{array}{l} 6s^2t - 2st^2 \\ 4s^2t - 3st^2 \end{array} \}

To find the sum of these polynomials, we need to combine like terms. This involves adding or subtracting the coefficients of the same degree variables.

Combining Like Terms

To find the sum of the two polynomials, we will combine like terms. The first polynomial is $6s^2t - 2st^2$ , and the second polynomial is $4s^2t - 3st^2$ . We can start by identifying the like terms in both polynomials.

The like terms in the first polynomial are $6s^2t$ and $-2st^2$ . The like terms in the second polynomial are $4s^2t$ and $-3st^2$ .

Adding the Like Terms

Now that we have identified the like terms, we can add them together. To do this, we will add the coefficients of the same degree variables.

The sum of the like terms is:

$6s^2t + 4s^2t = 10s^2t$

$-2st^2 - 3st^2 = -5st^2$

Determining the Nature of the Sum

Now that we have found the sum of the two polynomials, we can determine its nature. The sum is $10s^2t - 5st^2$ . This is a polynomial with two terms, where the first term is $10s^2t$ and the second term is $-5st^2$ .

Is the Sum a Binomial or a Trinomial?

A binomial is a polynomial with two terms, while a trinomial is a polynomial with three terms. Since the sum of the two polynomials has two terms, it is a binomial.

Is the Degree of the Sum 2 or 3?

The degree of a polynomial is the highest degree of any term in the polynomial. In the sum $10s^2t - 5st^2$ , the highest degree is 2. Therefore, the degree of the sum is 2.

Conclusion

In conclusion, the sum of the two polynomials is a binomial with a degree of 2. This is because the sum has two terms, and the highest degree of any term in the sum is 2.

Final Answer

The final answer is A. The sum is a binomial with a degree of 2.

Discussion

The discussion of this problem involves understanding the properties of polynomials and how to combine like terms. It also involves determining the nature of the sum, including whether it is a binomial or a trinomial, and the degree of the sum.

Importance of Understanding Polynomials

Understanding polynomials is crucial in various mathematical applications, including algebra, calculus, and engineering. Polynomials are used to model real-world phenomena, and understanding their properties and behavior is essential in solving problems and making predictions.

Real-World Applications of Polynomials

Polynomials have numerous real-world applications, including:

Modeling population growth and decline
Describing the motion of objects
Analyzing data and making predictions
Solving optimization problems

Conclusion

In conclusion, the sum of the two polynomials is a binomial with a degree of 2. This is because the sum has two terms, and the highest degree of any term in the sum is 2. Understanding polynomials and their properties is crucial in various mathematical applications, and it has numerous real-world applications.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Engineering Mathematics" by K.A. Stroud

Keywords

Polynomials
Binomial
Trinomial
Degree of a polynomial
Like terms
Combining like terms
Real-world applications of polynomials

Introduction

Polynomials are a fundamental concept in mathematics, and understanding their properties and behavior is crucial in various mathematical applications. In this article, we will answer some frequently asked questions about polynomials and their properties.

Q: What is a polynomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be written in the form:

$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are coefficients, and $x$ is the variable.

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest degree of any term in the polynomial. For example, in the polynomial $2x^3 + 3x^2 - 4x + 1$ , the degree is 3.

Q: What is a binomial?

A: A binomial is a polynomial with two terms. For example, $2x + 3$ is a binomial.

Q: What is a trinomial?

A: A trinomial is a polynomial with three terms. For example, $2x^2 + 3x - 4$ is a trinomial.

Q: How do I combine like terms in a polynomial?

A: To combine like terms in a polynomial, you need to add or subtract the coefficients of the same degree variables. For example, in the polynomial $2x^2 + 3x^2 - 4x$ , the like terms are $2x^2$ and $3x^2$ . Combining these terms gives $5x^2 - 4x$ .

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

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

Q: Can a polynomial have a negative degree?

A: No, a polynomial cannot have a negative degree. The degree of a polynomial is always a non-negative integer.

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

A: To determine the degree of a polynomial with multiple variables, you need to find the highest degree of any term in the polynomial. For example, in the polynomial $2x^2y + 3xy^2 - 4xy$ , the highest degree is 2.

Q: Can a polynomial have a variable with a fractional exponent?

A: No, a polynomial cannot have a variable with a fractional exponent. The exponent of a variable in a polynomial must be a non-negative integer.

Q: How do I factor a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials. There are various methods for factoring polynomials, including the use of the distributive property, the use of the quadratic formula, and the use of synthetic division.

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

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A rational function, on the other hand, is an expression that consists of a polynomial divided by another polynomial.

Q: Can a polynomial have a variable with a negative exponent?

A: No, a polynomial cannot have a variable with a negative exponent. The exponent of a variable in a polynomial must be a non-negative integer.

Conclusion

In conclusion, understanding polynomials and their properties is crucial in various mathematical applications. By answering these frequently asked questions, we hope to have provided a better understanding of polynomials and their behavior.

Final Answer

The final answer is that polynomials are a fundamental concept in mathematics, and understanding their properties and behavior is crucial in various mathematical applications.

Discussion

The discussion of this article involves understanding the properties of polynomials and how to combine like terms, determine the degree of a polynomial, and factor a polynomial. It also involves understanding the difference between a polynomial and an expression, a polynomial and a rational function, and a polynomial with a variable and a polynomial with a constant.

Importance of Understanding Polynomials

Real-World Applications of Polynomials

Polynomials have numerous real-world applications, including:

Modeling population growth and decline
Describing the motion of objects
Analyzing data and making predictions
Solving optimization problems

Conclusion

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Engineering Mathematics" by K.A. Stroud

Keywords

Polynomials
Binomial
Trinomial
Degree of a polynomial
Like terms
Combining like terms
Real-world applications of polynomials