Subtracting PolynomialsFind: $\left(4x^2y^3 + 2xy^2 - 2y\right) - \left(-7x^2y^3 + 6xy^2 - 2y\right$\]Place The Correct Coefficients In The Difference.$\square \, X^2y^3 + \square \, Xy^2 + \square \, Y$

Feb 26, 2025 by ADMIN 204 views

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Introduction

Subtracting polynomials is a fundamental operation in algebra that involves combining like terms to simplify expressions. In this article, we will explore the process of subtracting polynomials, including the rules and steps involved. We will also work through a specific example to illustrate the concept.

Rules for Subtracting Polynomials

When subtracting polynomials, we need to follow certain rules to ensure that we combine like terms correctly. Here are the key rules to keep in mind:

Like terms must be combined: When subtracting polynomials, we need to combine like terms, which are terms that have the same variable(s) raised to the same power.
Coefficients must be added or subtracted: When combining like terms, we add or subtract the coefficients of the terms.
Variables remain the same: The variables in the terms remain the same, and we only change the coefficients.

Steps for Subtracting Polynomials

Now that we have covered the rules for subtracting polynomials, let's go through the steps involved in the process:

Identify like terms: The first step is to identify like terms in the polynomials being subtracted. Like terms are terms that have the same variable(s) raised to the same power.
Combine like terms: Once we have identified like terms, we combine them by adding or subtracting the coefficients of the terms.
Simplify the expression: After combining like terms, we simplify the expression by removing any unnecessary terms.

Example: Subtracting Polynomials

Let's work through a specific example to illustrate the concept of subtracting polynomials. We will subtract the following polynomials:

$\left(4x^2y^3 + 2xy^2 - 2y\right) - \left(-7x^2y^3 + 6xy^2 - 2y\right)$

To subtract these polynomials, we need to follow the steps outlined above.

Step 1: Identify Like Terms

The first step is to identify like terms in the polynomials being subtracted. In this case, we have the following like terms:

$4x^2y^3$ and $-7x^2y^3$
$2xy^2$ and $6xy^2$
$-2y$ and $-2y$

Step 2: Combine Like Terms

Once we have identified like terms, we combine them by adding or subtracting the coefficients of the terms. In this case, we have the following combinations:

$4x^2y^3 - (-7x^2y^3) = 4x^2y^3 + 7x^2y^3 = 11x^2y^3$
$2xy^2 - 6xy^2 = -4xy^2$
$-2y - (-2y) = -2y + 2y = 0$

Step 3: Simplify the Expression

After combining like terms, we simplify the expression by removing any unnecessary terms. In this case, we have the following simplified expression:

$11x^2y^3 - 4xy^2$

Conclusion

Subtracting polynomials is a fundamental operation in algebra that involves combining like terms to simplify expressions. By following the rules and steps outlined above, we can subtract polynomials with ease. In this article, we worked through a specific example to illustrate the concept of subtracting polynomials.

Frequently Asked Questions

Here are some frequently asked questions about subtracting polynomials:

What are like terms?
- Like terms are terms that have the same variable(s) raised to the same power.
How do I combine like terms?
- To combine like terms, we add or subtract the coefficients of the terms.
What is the difference between adding and subtracting polynomials?
- The difference between adding and subtracting polynomials is that when we add polynomials, we combine like terms by adding the coefficients, whereas when we subtract polynomials, we combine like terms by subtracting the coefficients.

Final Thoughts

Subtracting polynomials is an essential operation in algebra that involves combining like terms to simplify expressions. By following the rules and steps outlined above, we can subtract polynomials with ease. Whether you are a student or a teacher, understanding how to subtract polynomials is crucial for success in algebra and beyond.

References

Here are some references that you may find helpful when learning about subtracting polynomials:

Additional Resources

Here are some additional resources that you may find helpful when learning about subtracting polynomials:

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Introduction

Subtracting polynomials is a fundamental operation in algebra that involves combining like terms to simplify expressions. In this article, we will answer some frequently asked questions about subtracting polynomials, including the rules and steps involved.

Q&A

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, $2x^2$ and $-3x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, we add or subtract the coefficients of the terms. For example, if we have the terms $2x^2$ and $-3x^2$ , we can combine them by adding the coefficients: $2x^2 - 3x^2 = -x^2$ .

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

A: The difference between adding and subtracting polynomials is that when we add polynomials, we combine like terms by adding the coefficients, whereas when we subtract polynomials, we combine like terms by subtracting the coefficients.

Q: How do I simplify an expression after combining like terms?

A: After combining like terms, we simplify the expression by removing any unnecessary terms. For example, if we have the expression $2x^2 - 3x^2 + 4x^2$ , we can simplify it by combining the like terms: $2x^2 - 3x^2 + 4x^2 = 3x^2$ .

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

A: No, you cannot subtract a polynomial from a non-polynomial expression. For example, you cannot subtract the polynomial $2x^2$ from the non-polynomial expression $3x + 4$ .

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

A: No, you cannot subtract a polynomial from a polynomial with a different variable. For example, you cannot subtract the polynomial $2x^2$ from the polynomial $3y^2$ .

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

A: When subtracting polynomials, we handle negative coefficients by changing the sign of the coefficient. For example, if we have the terms $2x^2$ and $-3x^2$ , we can combine them by adding the coefficients: $2x^2 - 3x^2 = -x^2$ .

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

A: Yes, you can subtract a polynomial from a polynomial with a different degree. For example, you can subtract the polynomial $2x^2$ from the polynomial $3x^3$ .