Subtract the coefficients of like terms

Oct 13, 2025 by magicalceylontours.com 11 views

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When we are given two polynomials, we can find their difference by subtracting one from the other. In this article, we will explore the concept of subtracting polynomials and provide a step-by-step guide on how to find the difference of two polynomials.

Understanding Polynomials

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are often represented by letters such as x, y, or z, and the coefficients are numbers that are multiplied with the variables. For example, the expression 2x^2 + 3x - 4 is a polynomial, where 2, 3, and -4 are the coefficients, and x is the variable.

Subtracting Polynomials

When we are given two polynomials, we can find their difference by subtracting one from the other. To do this, we need to follow the rules of subtraction, which are:

Subtract the coefficients of like terms.
Combine the like terms.

Step-by-Step Guide to Subtracting Polynomials

Let's consider the following example:

${ \left(-2x^3y^2 + 4x^2y^3 - 3xy^4\right) - \left(6x^4y - 5x^2y^3 - y^5\right) }$

To find the difference of these two polynomials, we need to follow the steps below:

Identify the like terms: The like terms in the given polynomials are the terms with the same variables and exponents. In this case, the like terms are -2x^3y2, 4x^2y3, -3xy^4, 6x^4y, -5x^2y3, and -y^5.
Subtract the coefficients of like terms: To subtract the coefficients of like terms, we need to subtract the coefficients of the corresponding terms. In this case, we need to subtract the coefficients of -2x^3y2, 4x^2y3, -3xy^4, 6x^4y, -5x^2y3, and -y^5.
Combine the like terms: After subtracting the coefficients of like terms, we need to combine the like terms. In this case, we need to combine the terms -2x^3y2, 4x^2y3, -3xy^4, 6x^4y, -5x^2y3, and -y^5.

Finding the Difference of Polynomials

Now that we have identified the like terms, subtracted the coefficients of like terms, and combined the like terms, we can find the difference of the two polynomials.

${ \left(-2x^3y^2 + 4x^2y^3 - 3xy^4\right) - \left(6x^4y - 5x^2y^3 - y^5\right) }$

${ = -2x^3y^2 + 4x^2y^3 - 3xy^4 - 6x^4y + 5x^2y^3 + y^5 }$

${ = -2x^3y^2 + 9x^2y^3 - 3xy^4 - 6x^4y + y^5 }$

Therefore, the difference of the two polynomials is:

${ -6x^4y - 2x^3y^2 + 9x^2y^3 - 3xy^4 + y^5 }$

Conclusion

In this article, we have explored the concept of subtracting polynomials and provided a step-by-step guide on how to find the difference of two polynomials. We have identified the like terms, subtracted the coefficients of like terms, and combined the like terms to find the difference of the two polynomials. The difference of the two polynomials is:

${ -6x^4y - 2x^3y^2 + 9x^2y^3 - 3xy^4 + y^5 }$

This is the correct answer to the given problem.

Answer Key

The correct answer to the given problem is:

A. ${-6x^4y - 2x^3y^2 + 9x^2y^3 - 3xy^4 + y^5}$

Discussion

This problem requires the student to understand the concept of subtracting polynomials and to apply the rules of subtraction to find the difference of two polynomials. The student needs to identify the like terms, subtract the coefficients of like terms, and combine the like terms to find the difference of the two polynomials.

Example Questions

Find the difference of the following two polynomials:

${ \left(2x^2y - 3xy^2 + 4y^3\right) - \left(x^2y - 2xy^2 + y^3\right) }$

Find the difference of the following two polynomials:

${ \left(3x^3y^2 - 2x^2y^3 + xy^4\right) - \left(4x^3y^2 - 3x^2y^3 + 2xy^4\right) }$

Solutions

The difference of the two polynomials is:

${ 2x^2y - 3xy^2 + 4y^3 - x^2y + 2xy^2 - y^3 }$

${ = x^2y + xy^2 + 3y^3 }$

The difference of the two polynomials is:

${ 3x^3y^2 - 2x^2y^3 + xy^4 - 4x^3y^2 + 3x^2y^3 - 2xy^4 }$

${ = -x^3y^2 + x^2y^3 - xy^4 }$

Conclusion

${ -6x^4y - 2x^3y^2 + 9x^2y^3 - 3xy^4 + y^5 }$

In this article, we will answer some frequently asked questions about subtracting polynomials. We will provide step-by-step explanations and examples to help you understand the concept of subtracting polynomials.

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

A: The main difference between subtracting polynomials and adding polynomials is the operation we are performing. When we add polynomials, we combine like terms by adding their coefficients. When we subtract polynomials, we combine like terms by subtracting their coefficients.

Q: How do I identify like terms in a polynomial?

A: To identify like terms in a polynomial, we need to look for terms with the same variables and exponents. For example, in the polynomial 2x^2 + 3x - 4, the like terms are 2x^2 and 3x.

Q: What is the rule for subtracting coefficients of like terms?

A: The rule for subtracting coefficients of like terms is to subtract the coefficient of the second term from the coefficient of the first term. For example, if we have the terms 2x^2 and 3x^2, we would subtract 3 from 2 to get -1x^2.

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

A: No, you cannot subtract a polynomial from a non-polynomial expression. Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Non-polynomial expressions, on the other hand, may contain other operations such as division or exponentiation.

Q: How do I simplify a polynomial after subtracting another polynomial?

A: To simplify a polynomial after subtracting another polynomial, we need to combine like terms by adding or subtracting their coefficients. For example, if we have the polynomial 2x^2 + 3x - 4 and we subtract the polynomial x^2 + 2x - 1, we would get:

2x^2 + 3x - 4 - (x^2 + 2x - 1)

= 2x^2 - x^2 + 3x - 2x - 4 + 1

= x^2 + x - 3

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 + 3x - 4 from the polynomial 2y^2 + 3y - 4, because the variables are different.

Q: How do I know if a polynomial is a difference of two polynomials?

A: To determine if a polynomial is a difference of two polynomials, we need to look for the presence of two or more terms with the same variables and exponents. If we can identify two or more terms with the same variables and exponents, we can rewrite the polynomial as the difference of two polynomials.

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 + 3x - 4 from the polynomial 4x^3 + 2x^2 + 3x - 1.

Q: How do I simplify a polynomial after subtracting another polynomial with a different degree?

A: To simplify a polynomial after subtracting another polynomial with a different degree, we need to combine like terms by adding or subtracting their coefficients. We also need to consider the degree of the resulting polynomial.

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

A: Yes, you can subtract a polynomial from a polynomial with a different coefficient. For example, you can subtract the polynomial 2x^2 + 3x - 4 from the polynomial 4x^2 + 3x - 1.

Q: How do I simplify a polynomial after subtracting another polynomial with a different coefficient?

A: To simplify a polynomial after subtracting another polynomial with a different coefficient, we need to combine like terms by adding or subtracting their coefficients.

Conclusion

In this article, we have answered some frequently asked questions about subtracting polynomials. We have provided step-by-step explanations and examples to help you understand the concept of subtracting polynomials. We hope this article has been helpful in clarifying any doubts you may have had about subtracting polynomials.