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) = \\]

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Introduction

Polynomials are a fundamental concept in algebra, and subtracting them is an essential operation in mathematics. In this article, we will explore the process of subtracting polynomials, including the rules and steps involved. We will also provide examples to illustrate the concept.

What are Polynomials?

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_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n, a_(n-1), ..., a_1, a_0 are constants, and x is the variable.

Subtracting Polynomials

Subtracting polynomials involves combining like terms and simplifying the expression. The rules for subtracting polynomials are as follows:

  • Combine like terms: Combine the coefficients of the same variables.
  • Simplify the expression: Simplify the resulting expression by combining like terms.

Example 1: Subtracting Two Polynomials

Let's consider the following example:

(59x+35y−512)−(49x+25y)\left(\frac{5}{9} x + \frac{3}{5} y - \frac{5}{12}\right) - \left(\frac{4}{9} x + \frac{2}{5} y\right)

To subtract these polynomials, we need to combine like terms and simplify the expression.

Step 1: Combine Like Terms

The first step is to combine the like terms in the two polynomials. In this case, we have:

  • 59x\frac{5}{9} x and −49x-\frac{4}{9} x (like terms)
  • 35y\frac{3}{5} y and −25y-\frac{2}{5} y (like terms)

We can combine these like terms as follows:

(59x−49x)+(35y−25y)\left(\frac{5}{9} x - \frac{4}{9} x\right) + \left(\frac{3}{5} y - \frac{2}{5} y\right)

Step 2: Simplify the Expression

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

(59x−49x)+(35y−25y)=19x+15y\left(\frac{5}{9} x - \frac{4}{9} x\right) + \left(\frac{3}{5} y - \frac{2}{5} y\right) = \frac{1}{9} x + \frac{1}{5} y

Step 3: Simplify the Constant Term

The final step is to simplify the constant term. In this case, we have:

−512−0=−512-\frac{5}{12} - 0 = -\frac{5}{12}

Therefore, the final result is:

19x+15y−512\frac{1}{9} x + \frac{1}{5} y - \frac{5}{12}

Conclusion

Subtracting polynomials involves combining like terms and simplifying the expression. By following the rules and steps outlined in this article, you can easily subtract polynomials and simplify the resulting expression.

Frequently Asked Questions

Q: What are the rules for subtracting polynomials?

A: The rules for subtracting polynomials are as follows:

  • Combine like terms: Combine the coefficients of the same variables.
  • Simplify the expression: Simplify the resulting expression by combining like terms.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the same variables.

Q: How do I simplify the expression?

A: To simplify the expression, you need to combine like terms and simplify the resulting expression.

Final Answer

The final answer is:

19x+15y−512\boxed{\frac{1}{9} x + \frac{1}{5} y - \frac{5}{12}}

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Introduction

Subtracting polynomials is an essential operation in mathematics, and it can be a bit tricky to understand at first. In this article, we will provide a Q&A section to help you better understand the concept of subtracting polynomials.

Q&A

Q: What are the rules for subtracting polynomials?

A: The rules for subtracting polynomials are as follows:

  • Combine like terms: Combine the coefficients of the same variables.
  • Simplify the expression: Simplify the resulting expression by combining like terms.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the same variables. For example, if you have two terms with the same variable, such as 2x and 3x, you can combine them by adding their coefficients: 2x + 3x = 5x.

Q: How do I simplify the expression?

A: To simplify the expression, you need to combine like terms and simplify the resulting expression. For example, if you have the expression 2x + 3x - 4x, you can simplify it by combining the like terms: 2x + 3x - 4x = x.

Q: What is the difference between subtracting polynomials and adding polynomials?

A: The main difference between subtracting polynomials and adding polynomials is that when you subtract polynomials, you need to combine like terms and simplify the expression, whereas when you add polynomials, you simply add the coefficients of the same variables.

Q: Can I subtract polynomials with different variables?

A: Yes, you can subtract polynomials with different variables. For example, if you have the expression 2x + 3y and you want to subtract 4x + 2y, you can do so by combining like terms: 2x + 3y - 4x - 2y = -2x + y.

Q: How do I handle negative coefficients when subtracting polynomials?

A: When subtracting polynomials, you need to handle negative coefficients by changing their sign. For example, if you have the expression -2x + 3y and you want to subtract 4x + 2y, you can do so by changing the sign of the negative coefficient: -2x + 3y - 4x - 2y = -6x + y.

Q: Can I subtract polynomials with fractions?

A: Yes, you can subtract polynomials with fractions. For example, if you have the expression 2/3x + 1/2y and you want to subtract 1/3x + 1/2y, you can do so by combining like terms: 2/3x + 1/2y - 1/3x - 1/2y = 1/6x.

Q: How do I handle polynomials with exponents when subtracting?

A: When subtracting polynomials with exponents, you need to combine like terms and simplify the expression. For example, if you have the expression 2x^2 + 3x and you want to subtract 4x^2 + 2x, you can do so by combining like terms: 2x^2 + 3x - 4x^2 - 2x = -2x^2 + x.

Conclusion

Subtracting polynomials can be a bit tricky, but with the right rules and techniques, you can easily subtract polynomials and simplify the resulting expression. We hope this Q&A section has helped you better understand the concept of subtracting polynomials.

Frequently Asked Questions

Q: What are some common mistakes to avoid when subtracting polynomials?

A: Some common mistakes to avoid when subtracting polynomials include:

  • Not combining like terms
  • Not simplifying the expression
  • Not handling negative coefficients correctly
  • Not handling fractions correctly
  • Not handling exponents correctly

Q: How do I practice subtracting polynomials?

A: To practice subtracting polynomials, you can try the following:

  • Start with simple expressions and gradually move on to more complex ones
  • Use online resources or worksheets to practice subtracting polynomials
  • Ask a teacher or tutor for help if you need it
  • Practice regularly to build your skills and confidence

Final Answer

The final answer is:

19x+15y−512\boxed{\frac{1}{9} x + \frac{1}{5} y - \frac{5}{12}}