Expanding the Quadratic Expression

Oct 13, 2025 by miodeya.com 47 views

Understanding the Product of a Quadratic Expression and its Implications on the Closure Property of Multiplication

In mathematics, the closure property of multiplication states that the product of any two real numbers is always a real number. This property is fundamental to the concept of multiplication and is used extensively in various mathematical operations. In this article, we will explore the product of the quadratic expression ${$ (x-9)^2$}$ and explain how it demonstrates the closure property of multiplication.

The Quadratic Expression

The given quadratic expression is ${$ (x-9)^2$}$. To find the product of this expression, we need to expand it using the formula ${$ (a-b)^2 = a^2 - 2ab + b^2$}$.

Expanding the Quadratic Expression

Using the formula, we can expand the quadratic expression as follows:

${$ (x-9)^2 = x^2 - 2 \cdot x \cdot 9 + 9^2$}$

Simplifying the expression, we get:

${$ (x-9)^2 = x^2 - 18x + 81$}$

The Product of the Quadratic Expression

Now that we have expanded the quadratic expression, we can find its product. However, the product of a quadratic expression is not a new expression, but rather the result of multiplying the expression by another expression. In this case, we will multiply the quadratic expression by 1, which is a real number.

Multiplying the Quadratic Expression by 1

To find the product of the quadratic expression and 1, we simply multiply the expression by 1:

${$ (x-9)^2 \cdot 1 = (x^2 - 18x + 81) \cdot 1$}$

Simplifying the expression, we get:

${$ x^2 - 18x + 81$}$

Demonstrating the Closure Property of Multiplication

The product of the quadratic expression and 1 is a polynomial, which is a mathematical expression consisting of variables and coefficients. The fact that the product is a polynomial demonstrates the closure property of multiplication, which states that the product of any two real numbers is always a real number.

In conclusion, the product of the quadratic expression ${$ (x-9)^2$}$ and 1 is ${$ x^2 - 18x + 81$}$, which is a polynomial. This demonstrates the closure property of multiplication, which is a fundamental concept in mathematics. The closure property of multiplication states that the product of any two real numbers is always a real number, and this property is used extensively in various mathematical operations.

Based on the above explanation, the correct answer to the question is:

A. ${$ x^2 - 81$}$; is a polynomial

This is because the product of the quadratic expression and 1 is indeed a polynomial, which is ${$ x^2 - 18x + 81$}$.

The closure property of multiplication is a fundamental concept in mathematics that states that the product of any two real numbers is always a real number.
The product of a quadratic expression and 1 is a polynomial, which demonstrates the closure property of multiplication.
The quadratic expression ${$ (x-9)^2$}$ can be expanded using the formula ${$ (a-b)^2 = a^2 - 2ab + b^2$}$.
The product of the quadratic expression and 1 is ${$ x^2 - 18x + 81$}$, which is a polynomial.
Q&A: Understanding the Product of a Quadratic Expression and its Implications on the Closure Property of Multiplication

In our previous article, we explored the product of the quadratic expression ${$ (x-9)^2$}$ and explained how it demonstrates the closure property of multiplication. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the closure property of multiplication?

A: The closure property of multiplication states that the product of any two real numbers is always a real number. This means that when you multiply two real numbers together, the result is always a real number.

Q: How does the product of a quadratic expression demonstrate the closure property of multiplication?

A: The product of a quadratic expression and 1 is a polynomial, which demonstrates the closure property of multiplication. This is because the product of any two real numbers is always a real number, and in this case, the product of the quadratic expression and 1 is a polynomial.

Q: What is a polynomial?

A: A polynomial is a mathematical expression consisting of variables and coefficients. It is a sum of terms, where each term is a product of a variable and a coefficient.

Q: Can you give an example of a polynomial?

A: Yes, an example of a polynomial is ${$ x^2 - 18x + 81$}$. This is a polynomial because it is a sum of terms, where each term is a product of a variable and a coefficient.

Q: How do you expand a quadratic expression?

A: To expand a quadratic expression, you can use the formula ${$ (a-b)^2 = a^2 - 2ab + b^2$}$. This formula allows you to expand the expression and simplify it.

Q: Can you give an example of expanding a quadratic expression?

A: Yes, an example of expanding a quadratic expression is:

${$ (x-9)^2 = x^2 - 2 \cdot x \cdot 9 + 9^2$}$

Simplifying the expression, we get:

${$ (x-9)^2 = x^2 - 18x + 81$}$

Q: What is the product of the quadratic expression and 1?

A: The product of the quadratic expression and 1 is ${$ x^2 - 18x + 81$}$. This is a polynomial, which demonstrates the closure property of multiplication.

Q: Can you explain the concept of the closure property of multiplication in simple terms?

A: The closure property of multiplication is a simple concept that states that when you multiply two real numbers together, the result is always a real number. This means that you can always multiply two real numbers together and get a real number as a result.

In conclusion, the product of a quadratic expression and 1 is a polynomial, which demonstrates the closure property of multiplication. This concept is fundamental to mathematics and is used extensively in various mathematical operations. We hope that this article has helped to clarify any questions you may have had about this topic.

The closure property of multiplication is a fundamental concept in mathematics that states that the product of any two real numbers is always a real number.
The product of a quadratic expression and 1 is a polynomial, which demonstrates the closure property of multiplication.
The quadratic expression ${$ (x-9)^2$}$ can be expanded using the formula ${$ (a-b)^2 = a^2 - 2ab + b^2$}$.
The product of the quadratic expression and 1 is ${$ x^2 - 18x + 81$}$, which is a polynomial.