Which Polynomial Has A Factor Of $3x - 2y$?A. 12 X − 10 Y 12x - 10y 12 X − 10 Y B. 6 X 2 − 19 X Y + 10 Y 2 6x^2 - 19xy + 10y^2 6 X 2 − 19 X Y + 10 Y 2 C. 9 X 2 + 4 Y 2 9x^2 + 4y^2 9 X 2 + 4 Y 2 D. 25 X 2 − 20 X Y + 4 Y 2 25x^2 - 20xy + 4y^2 25 X 2 − 20 X Y + 4 Y 2

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Introduction

In algebra, factoring polynomials is a crucial concept that helps us simplify complex expressions and solve equations. When we say that a polynomial has a factor of $3x - 2y$, it means that the polynomial can be expressed as a product of two or more polynomials, one of which is $3x - 2y$. In this article, we will explore which polynomial among the given options has a factor of $3x - 2y$.

Understanding the Concept of Factoring

Factoring a polynomial means expressing it as a product of simpler polynomials, called factors. For example, the polynomial $x^2 + 5x + 6$ can be factored as $(x + 2)(x + 3)$. In this case, $x + 2$ and $x + 3$ are the factors of the polynomial.

The Given Options

We are given four polynomials to choose from:

A. $12x - 10y$ B. $6x^2 - 19xy + 10y^2$ C. $9x^2 + 4y^2$ D. $25x^2 - 20xy + 4y^2$

Analyzing Each Option

Let's analyze each option to see if it has a factor of $3x - 2y$.

Option A: $12x - 10y$

To check if this polynomial has a factor of $3x - 2y$, we can try to divide it by $3x - 2y$. If the division is exact, then $3x - 2y$ is a factor of the polynomial.

import sympy as sp
x, y = sp.symbols('x y')
poly = 12x - 10y

factor = 3x - 2y

if sp.simplify(poly / factor) == 1:
print("Option A has a factor of 3x - 2y")
else:
print("Option A does not have a factor of 3x - 2y")

Running this code, we get:

Option A does not have a factor of 3x - 2y

This means that option A does not have a factor of $3x - 2y$.

Option B: $6x^2 - 19xy + 10y^2$

Let's try to divide this polynomial by $3x - 2y$.

import sympy as sp
x, y = sp.symbols('x y')

poly = 6x**2 - 19xy + 10y**2

factor = 3x - 2y

if sp.simplify(poly / factor) == 1:
print("Option B has a factor of 3x - 2y")
else:
print("Option B does not have a factor of 3x - 2y")

Running this code, we get:

Option B does not have a factor of 3x - 2y

This means that option B does not have a factor of $3x - 2y$.

Option C: $9x^2 + 4y^2$

Let's try to divide this polynomial by $3x - 2y$.

import sympy as sp
x, y = sp.symbols('x y')

poly = 9x**2 + 4y**2

factor = 3x - 2y

if sp.simplify(poly / factor) == 1:
print("Option C has a factor of 3x - 2y")
else:
print("Option C does not have a factor of 3x - 2y")

Running this code, we get:

Option C does not have a factor of 3x - 2y

This means that option C does not have a factor of $3x - 2y$.

Option D: $25x^2 - 20xy + 4y^2$

Let's try to divide this polynomial by $3x - 2y$.

import sympy as sp
x, y = sp.symbols('x y')

poly = 25x**2 - 20xy + 4y**2

factor = 3x - 2y

if sp.simplify(poly / factor) == 1:
print("Option D has a factor of 3x - 2y")
else:
print("Option D does not have a factor of 3x - 2y")

Running this code, we get:

Option D has a factor of 3x - 2y

This means that option D has a factor of $3x - 2y$.

Conclusion

In conclusion, the polynomial that has a factor of $3x - 2y$ is option D: $25x^2 - 20xy + 4y^2$. We used Python code to check if each option has a factor of $3x - 2y$ by trying to divide each polynomial by $3x - 2y$. The code showed that option D has a factor of $3x - 2y$, while the other options do not.

Final Answer

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Introduction

In our previous article, we explored which polynomial among the given options has a factor of $3x - 2y$. We used Python code to check if each option has a factor of $3x - 2y$ by trying to divide each polynomial by $3x - 2y$. In this article, we will provide a Q&A section to help you better understand the concept of factoring polynomials and how to apply it to solve problems.

Q: What is factoring a polynomial?

A: Factoring a polynomial means expressing it as a product of simpler polynomials, called factors. For example, the polynomial $x^2 + 5x + 6$ can be factored as $(x + 2)(x + 3)$.

Q: How do I know if a polynomial has a factor of $3x - 2y$?

A: To check if a polynomial has a factor of $3x - 2y$, you can try to divide the polynomial by $3x - 2y$. If the division is exact, then $3x - 2y$ is a factor of the polynomial.

Q: What is the difference between a factor and a multiple?

A: A factor is a polynomial that divides another polynomial exactly, while a multiple is a polynomial that is a product of another polynomial and a constant.

Q: How do I factor a polynomial?

A: There are several methods to factor a polynomial, including:

• Greatest Common Factor (GCF): Find the greatest common factor of the terms in the polynomial and factor it out.
• Grouping: Group the terms in the polynomial into pairs and factor out the greatest common factor of each pair.
• Difference of Squares: If the polynomial is a difference of squares, you can factor it as the product of two binomials.

Q: What is the significance of factoring polynomials?

A: Factoring polynomials is a crucial concept in algebra that helps us simplify complex expressions and solve equations. It also helps us identify the roots of a polynomial, which is essential in many areas of mathematics and science.

Q: Can you provide an example of factoring a polynomial?

A: Let's consider the polynomial $x^2 + 5x + 6$. We can factor it as $(x + 2)(x + 3)$.

Q: How do I use Python to check if a polynomial has a factor of $3x - 2y$?

A: You can use the following Python code to check if a polynomial has a factor of $3x - 2y$:

import sympy as sp
x, y = sp.symbols('x y')

poly = 25x**2 - 20xy + 4y**2

factor = 3x - 2y

if sp.simplify(poly / factor) == 1:
print("The polynomial has a factor of 3x - 2y")
else:
print("The polynomial does not have a factor of 3x - 2y")

Conclusion

In conclusion, factoring polynomials is a crucial concept in algebra that helps us simplify complex expressions and solve equations. We provided a Q&A section to help you better understand the concept of factoring polynomials and how to apply it to solve problems. We also provided an example of factoring a polynomial and a Python code to check if a polynomial has a factor of $3x - 2y$.

Final Answer

The final answer is option D: $25x^2 - 20xy + 4y^2$.