Subtract The Polynomials:$\[ \left(\frac{5}{9} X + \frac{3}{5} Y - \frac{5}{12}\right) - \left(\frac{4}{9} X + \frac{2}{5} Y\right) = \square \\]

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Introduction

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Polynomial subtraction is a fundamental concept in algebra, and it's essential to understand how to perform this operation to solve various mathematical problems. In this article, we will focus on subtracting two polynomials, which involves combining like terms and simplifying the resulting expression.

What are Polynomials?

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A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be written in various forms, including monomials, binomials, and trinomials. In this article, we will be dealing with polynomials in the form of:

ax + by + c

where a, b, and c are constants, and x and y are variables.

Subtracting Polynomials: A Step-by-Step Guide

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To subtract two polynomials, we need to follow these steps:

1. Identify like terms: Like terms are terms that have the same variable(s) raised to the same power. In the given problem, we have two polynomials:

   $\left(\frac{5}{9} x + \frac{3}{5} y - \frac{5}{12}\right) - \left(\frac{4}{9} x + \frac{2}{5} y\right)$

   We need to identify the like terms in both polynomials.

2. Combine like terms: Once we have identified the like terms, we can combine them by adding or subtracting their coefficients. In this case, we have two like terms:

   $\frac{5}{9} x$ and $\frac{4}{9} x$

   We can combine these terms by subtracting their coefficients:

   $\frac{5}{9} x - \frac{4}{9} x = \frac{1}{9} x$

3. Simplify the resulting expression: After combining like terms, we need to simplify the resulting expression by combining any remaining like terms.

Subtracting the Polynomials

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Now that we have identified the like terms and combined them, we can subtract the polynomials:

$\left(\frac{5}{9} x + \frac{3}{5} y - \frac{5}{12}\right) - \left(\frac{4}{9} x + \frac{2}{5} y\right)$

$= \frac{5}{9} x - \frac{4}{9} x + \frac{3}{5} y - \frac{2}{5} y - \frac{5}{12}$

$= \frac{1}{9} x + \frac{1}{5} y - \frac{5}{12}$

Conclusion

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Subtracting polynomials involves identifying like terms, combining them, and simplifying the resulting expression. By following these steps, we can perform polynomial subtraction with ease. In this article, we have subtracted two polynomials and simplified the resulting expression.

Frequently Asked Questions

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Q: What are like terms in polynomials?

A: Like terms are terms that have the same variable(s) raised to the same power.

Q: How do I combine like terms in polynomials?

A: To combine like terms, add or subtract their coefficients.

Q: What is the resulting expression after subtracting the polynomials?

A: The resulting expression is $\frac{1}{9} x + \frac{1}{5} y - \frac{5}{12}$.

Final Answer

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The final answer is $\boxed{\frac{1}{9} x + \frac{1}{5} y - \frac{5}{12}}$.

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Introduction

---

Polynomial subtraction is a fundamental concept in algebra, and it's essential to understand how to perform this operation to solve various mathematical problems. In this article, we will focus on providing answers to frequently asked questions related to polynomial subtraction.

Q&A: Polynomial Subtraction

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Q: What are like terms in polynomials?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, in the polynomial $2x + 3y + 4$, the like terms are $2x$ and $3y$.

Q: How do I identify like terms in polynomials?

A: To identify like terms, look for terms that have the same variable(s) raised to the same power. For example, in the polynomial $2x + 3y + 4x$, the like terms are $2x$ and $4x$.

Q: How do I combine like terms in polynomials?

A: To combine like terms, add or subtract their coefficients. For example, in the polynomial $2x + 3y + 4x$, the like terms $2x$ and $4x$ can be combined by adding their coefficients:

$2x + 4x = 6x$

Q: What is the order of operations for polynomial subtraction?

A: The order of operations for polynomial subtraction is:

1. Identify like terms: Identify the like terms in both polynomials.
2. Combine like terms: Combine the like terms by adding or subtracting their coefficients.
3. Simplify the resulting expression: Simplify the resulting expression by combining any remaining like terms.

Q: Can I subtract a polynomial from a non-polynomial expression?

A: No, you cannot subtract a polynomial from a non-polynomial expression. Polynomial subtraction is only possible when both expressions are polynomials.

Q: What is the resulting expression after subtracting two polynomials?

A: The resulting expression after subtracting two polynomials is a polynomial that contains the combined like terms and any remaining terms.

Q: Can I subtract a polynomial from a polynomial with a higher degree?

A: Yes, you can subtract a polynomial from a polynomial with a higher degree. However, the resulting expression will have a degree equal to the highest degree of the two polynomials.

Examples of Polynomial Subtraction

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Example 1: Subtracting two polynomials with like terms

Subtract the polynomials:

$\left(2x + 3y + 4\right) - \left(x + 2y + 3\right)$

Solution:

$= 2x - x + 3y - 2y + 4 - 3$

$= x + y + 1$

Example 2: Subtracting two polynomials with unlike terms

Subtract the polynomials:

$\left(2x + 3y + 4\right) - \left(x + 2y + 3z\right)$

Solution:

$= 2x - x + 3y - 2y + 4 - 3z$

$= x + y - 3z + 1$

Conclusion

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Polynomial subtraction is a fundamental concept in algebra, and it's essential to understand how to perform this operation to solve various mathematical problems. By following the steps outlined in this article, you can perform polynomial subtraction with ease.

Final Answer

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The final answer is $\boxed{1}$, which represents the number of steps required to perform polynomial subtraction.