Subtract: 3 X Y + 5 Y Z − 7 Z X 3xy + 5yz - 7zx 3 X Y + 5 Yz − 7 Z X From 5 X Y − 2 Y Z − 2 Z X + 10 X Y Z 5xy - 2yz - 2zx + 10xyz 5 X Y − 2 Yz − 2 Z X + 10 X Yz .

$$$$

Introduction

In algebra, subtraction is a fundamental operation that involves finding the difference between two or more polynomials. When subtracting polynomials, we need to follow a specific set of rules to ensure that the resulting expression is simplified and accurate. In this article, we will learn how to subtract the polynomial $3xy + 5yz - 7zx$ from the polynomial $5xy - 2yz - 2zx + 10xyz$.

Understanding Polynomials

Before we proceed with the subtraction, let's first understand what polynomials are. A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be written in the form of $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n, a_{n-1}, \ldots, a_1, a_0$ are coefficients and $x$ is the variable.

Subtracting Polynomials

To subtract polynomials, we need to follow the same rules as when adding polynomials. We need to combine like terms, which are terms that have the same variable and exponent. When subtracting polynomials, we need to change the sign of each term in the second polynomial and then combine like terms.

Step-by-Step Subtraction

Let's now proceed with the step-by-step subtraction of the polynomial $3xy + 5yz - 7zx$ from the polynomial $5xy - 2yz - 2zx + 10xyz$.

Step 1: Write down the two polynomials

The two polynomials are:

$5xy - 2yz - 2zx + 10xyz$

$3xy + 5yz - 7zx$

Step 2: Change the sign of each term in the second polynomial

To subtract the second polynomial from the first polynomial, we need to change the sign of each term in the second polynomial. This gives us:

$-3xy - 5yz + 7zx$

Step 3: Combine like terms

Now, we need to combine like terms. The like terms in this case are the terms with the same variable and exponent. We can combine the terms as follows:

$-3xy - 5yz + 7zx = -3xy - 5yz + 7zx$

Step 4: Simplify the expression

The expression is already simplified, so we don't need to do anything further.

Final Answer

The final answer is:

$5xy - 2yz - 2zx + 10xyz - 3xy - 5yz + 7zx$

Simplifying the Expression

To simplify the expression, we need to combine like terms. The like terms in this case are the terms with the same variable and exponent. We can combine the terms as follows:

$5xy - 3xy = 2xy$

$-2yz - 5yz = -7yz$

$-2zx + 7zx = 5zx$

$10xyz$ remains the same.

Final Simplified Expression

The final simplified expression is:

$2xy - 7yz + 5zx + 10xyz$

Conclusion

In this article, we learned how to subtract the polynomial $3xy + 5yz - 7zx$ from the polynomial $5xy - 2yz - 2zx + 10xyz$. We followed the step-by-step process of changing the sign of each term in the second polynomial and combining like terms to simplify the expression. The final simplified expression is $2xy - 7yz + 5zx + 10xyz$.

Frequently Asked Questions

• What is the difference between adding and subtracting polynomials? Adding and subtracting polynomials involve combining like terms, but the sign of each term is changed when subtracting polynomials.
• How do I simplify a polynomial expression? To simplify a polynomial expression, you need to combine like terms by adding or subtracting the coefficients of the terms with the same variable and exponent.
• What is the final answer to the subtraction problem? The final answer to the subtraction problem is $2xy - 7yz + 5zx + 10xyz$.

Further Reading

If you want to learn more about polynomials and algebra, here are some recommended resources:

• Khan Academy: Algebra
• MIT OpenCourseWare: Algebra
• Wolfram MathWorld: Polynomials

References

• "Algebra" by Michael Artin
• "Polynomials" by David M. Burton
• "Algebra and Trigonometry" by James Stewart
  

Introduction

In our previous article, we learned how to subtract the polynomial $3xy + 5yz - 7zx$ from the polynomial $5xy - 2yz - 2zx + 10xyz$. In this article, we will answer some frequently asked questions about subtracting polynomials.

Q&A

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

A: The difference between adding and subtracting polynomials is that when subtracting polynomials, we need to change the sign of each term in the second polynomial. This is in contrast to adding polynomials, where we simply combine like terms.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you need to combine like terms by adding or subtracting the coefficients of the terms with the same variable and exponent.

Q: What is the final answer to the subtraction problem?

A: The final answer to the subtraction problem is $2xy - 7yz + 5zx + 10xyz$.

Q: Can I subtract a polynomial from a polynomial with a different variable?

A: Yes, you can subtract a polynomial from a polynomial with a different variable. However, you will need to combine like terms carefully, as the variables may have different exponents.

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

A: When subtracting polynomials, you need to change the sign of each term in the second polynomial. This means that if a term has a negative coefficient, you will need to change the sign of the entire term.

Q: Can I subtract a polynomial from a polynomial with a constant term?

A: Yes, you can subtract a polynomial from a polynomial with a constant term. However, you will need to combine like terms carefully, as the constant term may have a different sign.

Q: How do I handle fractions when subtracting polynomials?

A: When subtracting polynomials, you need to change the sign of each term in the second polynomial. This means that if a term has a fraction, you will need to change the sign of the entire term.

Q: Can I subtract a polynomial from a polynomial with a binomial term?

A: Yes, you can subtract a polynomial from a polynomial with a binomial term. However, you will need to combine like terms carefully, as the binomial term may have a different sign.

Q: How do I handle exponents when subtracting polynomials?

A: When subtracting polynomials, you need to combine like terms by adding or subtracting the coefficients of the terms with the same variable and exponent.

Tips and Tricks

• When subtracting polynomials, make sure to change the sign of each term in the second polynomial.
• Combine like terms carefully, as the variables may have different exponents.
• Handle negative coefficients, fractions, and binomial terms carefully when subtracting polynomials.
• Use the distributive property to simplify polynomial expressions.

Conclusion

In this article, we answered some frequently asked questions about subtracting polynomials. We covered topics such as the difference between adding and subtracting polynomials, simplifying polynomial expressions, and handling negative coefficients, fractions, and binomial terms. We also provided some tips and tricks for subtracting polynomials.

Frequently Asked Questions

• What is the difference between adding and subtracting polynomials?
• How do I simplify a polynomial expression?
• What is the final answer to the subtraction problem?
• Can I subtract a polynomial from a polynomial with a different variable?
• How do I handle negative coefficients when subtracting polynomials?
• Can I subtract a polynomial from a polynomial with a constant term?
• How do I handle fractions when subtracting polynomials?
• Can I subtract a polynomial from a polynomial with a binomial term?
• How do I handle exponents when subtracting polynomials?

Further Reading

If you want to learn more about polynomials and algebra, here are some recommended resources:

• Khan Academy: Algebra
• MIT OpenCourseWare: Algebra
• Wolfram MathWorld: Polynomials

References

• "Algebra" by Michael Artin
• "Polynomials" by David M. Burton
• "Algebra and Trigonometry" by James Stewart