Select The Correct Answer From Each Drop-down Menu.Find The Sum Of 5 X 2 Y 5x^2y 5 X 2 Y And ( 2 X Y 2 + X 2 Y (2xy^2 + X^2y ( 2 X Y 2 + X 2 Y ].The Sum Of 5 X 2 Y 5x^2y 5 X 2 Y And ( 2 X Y 2 + X 2 Y (2xy^2 + X^2y ( 2 X Y 2 + X 2 Y ] Is 6 X 2 Y + 2 X Y 2 6x^2y + 2xy^2 6 X 2 Y + 2 X Y 2 , Which Is A- □ \square □ Monomial-

Introduction

Algebraic expressions are a fundamental concept in mathematics, and solving them requires a clear understanding of the rules and operations involved. In this article, we will focus on finding the sum of two algebraic expressions, $5x^2y$ and $(2xy^2 + x^2y)$. We will break down the solution step by step, highlighting the key concepts and techniques used to arrive at the final answer.

Understanding Algebraic Expressions

Before we dive into the solution, let's take a moment to understand what algebraic expressions are. An algebraic expression is a combination of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division. Algebraic expressions can be represented using variables, which are letters or symbols that represent unknown values.

The Problem

The problem we are trying to solve is to find the sum of two algebraic expressions:

$5x^2y$ and $(2xy^2 + x^2y)$

Step 1: Distributive Property

To find the sum of the two expressions, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. In this case, we can distribute the $x^2y$ term to both expressions inside the parentheses.

(2xy^2 + x^2y) = 2xy^2 + x^2y

Step 2: Combine Like Terms

Now that we have distributed the $x^2y$ term, we can combine like terms. Like terms are terms that have the same variable(s) raised to the same power. In this case, we have two terms with the variable $x^2y$:

$5x^2y$ and $x^2y$

We can combine these two terms by adding their coefficients:

5x^2y + x^2y = 6x^2y

Step 3: Simplify the Expression

Now that we have combined like terms, we can simplify the expression by combining the remaining terms:

$2xy^2 + 6x^2y$

We can rewrite this expression as:

$6x^2y + 2xy^2$

Conclusion

In conclusion, the sum of $5x^2y$ and $(2xy^2 + x^2y)$ is $6x^2y + 2xy^2$. This expression is a binomial, which is a polynomial with two terms.

What is a Binomial?

A binomial is a polynomial with two terms. It is a type of algebraic expression that can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is a variable.

Why is it Important to Understand Binomials?

Understanding binomials is important because they are used extensively in algebra and other areas of mathematics. Binomials are used to represent a wide range of mathematical concepts, including quadratic equations, polynomial functions, and statistical models.

Real-World Applications of Binomials

Binomials have numerous real-world applications, including:

• Quadratic Equations: Binomials are used to represent quadratic equations, which are used to model a wide range of phenomena, including the motion of objects, the growth of populations, and the behavior of electrical circuits.
• Polynomial Functions: Binomials are used to represent polynomial functions, which are used to model a wide range of phenomena, including the growth of populations, the behavior of electrical circuits, and the motion of objects.
• Statistical Models: Binomials are used to represent statistical models, which are used to analyze and interpret data in a wide range of fields, including medicine, economics, and social sciences.

Conclusion

$$$$$$

Q: What is a binomial?

A: A binomial is a polynomial with two terms. It is a type of algebraic expression that can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is a variable.

Q: What are some examples of binomials?

A: Some examples of binomials include:

• $2x + 3$
• $x^2 + 4x$
• $3y - 2$

Q: How do I add binomials?

A: To add binomials, you need to combine like terms. Like terms are terms that have the same variable(s) raised to the same power. For example, if you have the binomials $2x + 3$ and $x + 4$, you can add them by combining the like terms:

$2x + 3 + x + 4 = 3x + 7$

Q: How do I subtract binomials?

A: To subtract binomials, you need to combine like terms. Like terms are terms that have the same variable(s) raised to the same power. For example, if you have the binomials $2x + 3$ and $x + 4$, you can subtract them by combining the like terms:

$2x + 3 - (x + 4) = 2x + 3 - x - 4 = x - 1$

Q: Can I multiply binomials?

A: Yes, you can multiply binomials. To multiply binomials, you need to use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. For example, if you have the binomials $2x + 3$ and $x + 4$, you can multiply them by using the distributive property:

$(2x + 3)(x + 4) = 2x(x + 4) + 3(x + 4) = 2x^2 + 8x + 3x + 12 = 2x^2 + 11x + 12$

Q: Can I divide binomials?

A: Yes, you can divide binomials. To divide binomials, you need to use the quotient rule, which states that for any real numbers $a$, $b$, and $c$, $\frac{a}{b} = \frac{ac}{bc}$. For example, if you have the binomials $2x + 3$ and $x + 4$, you can divide them by using the quotient rule:

$\frac{2x + 3}{x + 4} = \frac{(2x + 3)(x + 4)}{(x + 4)(x + 4)} = \frac{2x^2 + 8x + 3x + 12}{x^2 + 8x + 16} = \frac{2x^2 + 11x + 12}{x^2 + 8x + 16}$

Q: What are some real-world applications of binomials?

A: Binomials have numerous real-world applications, including:

• Quadratic Equations: Binomials are used to represent quadratic equations, which are used to model a wide range of phenomena, including the motion of objects, the growth of populations, and the behavior of electrical circuits.
• Polynomial Functions: Binomials are used to represent polynomial functions, which are used to model a wide range of phenomena, including the growth of populations, the behavior of electrical circuits, and the motion of objects.
• Statistical Models: Binomials are used to represent statistical models, which are used to analyze and interpret data in a wide range of fields, including medicine, economics, and social sciences.

Conclusion

In conclusion, binomials are an important concept in algebra and other areas of mathematics. They have numerous real-world applications, including quadratic equations, polynomial functions, and statistical models. Understanding binomials is essential for solving a wide range of mathematical problems and for applying mathematical concepts to real-world situations.