Subtract { [4p^2 - 9p + 11]$}$ From { [p^2 - 5p + 4]$}$.Your Answer Should Be A Polynomial In Standard Form. { \square$}$

Mar 13, 2025 by ADMIN 122 views

Subtracting Polynomials: A Step-by-Step Guide

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Introduction

In algebra, polynomials are mathematical expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When subtracting polynomials, we need to follow a specific set of rules to ensure that the resulting expression is in standard form. In this article, we will explore the process of subtracting polynomials and provide a step-by-step guide on how to do it.

What are Polynomials?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are usually 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 + 3y - 4 is a polynomial.

Subtracting Polynomials

When subtracting polynomials, we need to follow the same rules as when adding polynomials. The main difference is that we need to change the sign of each term in the second polynomial. This is because subtraction is the opposite of addition, and we need to "undo" the addition by changing the sign of each term.

Step 1: Write Down the Polynomials

The first step in subtracting polynomials is to write down the two polynomials that we want to subtract. In this case, we have:

p^2 - 5p + 4

and

4p^2 - 9p + 11

Step 2: Change the Sign of Each Term

The next step is to change the sign of each term in the second polynomial. This means that we need to multiply each term by -1. So, the second polynomial becomes:

-4p^2 + 9p - 11

Step 3: Subtract the Terms

Now that we have changed the sign of each term in the second polynomial, we can subtract the terms. This means that we need to subtract each term in the second polynomial from the corresponding term in the first polynomial.

p^2 - 5p + 4

-4p^2 + 9p - 11

Subtracting the terms, we get:

-3p^2 + 4p - 7

Step 4: Simplify the Expression

The final step is to simplify the expression by combining like terms. In this case, we have:

-3p^2 + 4p - 7

This is the final answer, and it is a polynomial in standard form.

Conclusion

Subtracting polynomials is a straightforward process that involves changing the sign of each term in the second polynomial and then subtracting the terms. By following these steps, we can simplify complex expressions and arrive at a polynomial in standard form. Whether you are a student or a professional, understanding how to subtract polynomials is an essential skill that can help you solve a wide range of mathematical problems.

Example Problems

Here are a few example problems to help you practice subtracting polynomials:

Problem 1

Subtract the following polynomials:

3x^2 + 2x - 1

and

-2x^2 + 4x + 3

Solution

To subtract the polynomials, we need to change the sign of each term in the second polynomial. This means that we need to multiply each term by -1. So, the second polynomial becomes:

-2x^2 - 4x - 3

Now, we can subtract the terms:

3x^2 + 2x - 1

-2x^2 - 4x - 3

Subtracting the terms, we get:

5x^2 + 6x - 4

Problem 2

Subtract the following polynomials:

x^2 - 3x + 2

and

-2x^2 + 5x - 1

Solution

To subtract the polynomials, we need to change the sign of each term in the second polynomial. This means that we need to multiply each term by -1. So, the second polynomial becomes:

-2x^2 - 5x + 1

Now, we can subtract the terms:

x^2 - 3x + 2

-2x^2 - 5x + 1

Subtracting the terms, we get:

3x^2 + 2x - 3

Tips and Tricks

Here are a few tips and tricks to help you subtract polynomials:

Make sure to change the sign of each term in the second polynomial.
Subtract the terms by combining like terms.
Simplify the expression by combining like terms.
Check your work by plugging in values for the variables.

By following these tips and tricks, you can simplify complex expressions and arrive at a polynomial in standard form.

Common Mistakes

Here are a few common mistakes to avoid when subtracting polynomials:

Failing to change the sign of each term in the second polynomial.
Subtracting the terms incorrectly.
Failing to simplify the expression by combining like terms.
Not checking your work by plugging in values for the variables.

By avoiding these common mistakes, you can ensure that your answers are accurate and complete.

Final Thoughts

Subtracting polynomials is a fundamental concept in algebra that can help you solve a wide range of mathematical problems. By following the steps outlined in this article, you can simplify complex expressions and arrive at a polynomial in standard form. Whether you are a student or a professional, understanding how to subtract polynomials is an essential skill that can help you succeed in your mathematical endeavors.

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about subtracting polynomials.

Q: What is the first step in subtracting polynomials?

A: The first step in subtracting polynomials is to write down the two polynomials that you want to subtract.

Q: How do I change the sign of each term in the second polynomial?

A: To change the sign of each term in the second polynomial, you need to multiply each term by -1.

Q: What is the next step after changing the sign of each term in the second polynomial?

A: The next step is to subtract the terms by combining like terms.

Q: How do I simplify the expression by combining like terms?

A: To simplify the expression by combining like terms, you need to add or subtract the coefficients of the like terms.

Q: What is the final step in subtracting polynomials?

A: The final step is to check your work by plugging in values for the variables.

Common Questions

Here are some common questions that people ask about subtracting polynomials:

Q: What if the polynomials have different variables?

A: If the polynomials have different variables, you need to use the distributive property to expand the polynomials before subtracting them.

Q: What if the polynomials have different exponents?

A: If the polynomials have different exponents, you need to use the distributive property to expand the polynomials before subtracting them.

Q: What if the polynomials have a common factor?

A: If the polynomials have a common factor, you can factor out the common factor before subtracting the polynomials.

Advanced Questions

Here are some advanced questions that people ask about subtracting polynomials:

Q: What if the polynomials have complex coefficients?

A: If the polynomials have complex coefficients, you need to use the distributive property to expand the polynomials before subtracting them.

Q: What if the polynomials have rational coefficients?

A: If the polynomials have rational coefficients, you need to use the distributive property to expand the polynomials before subtracting them.

Q: What if the polynomials have polynomial coefficients?

A: If the polynomials have polynomial coefficients, you need to use the distributive property to expand the polynomials before subtracting them.

Conclusion

Example Problems

Here are a few example problems to help you practice subtracting polynomials:

Problem 1

Subtract the following polynomials:

3x^2 + 2x - 1

and

-2x^2 + 4x + 3

Solution

To subtract the polynomials, we need to change the sign of each term in the second polynomial. This means that we need to multiply each term by -1. So, the second polynomial becomes:

-2x^2 - 4x - 3

Now, we can subtract the terms:

3x^2 + 2x - 1

-2x^2 - 4x - 3

Subtracting the terms, we get:

5x^2 + 6x - 4

Problem 2

Subtract the following polynomials:

x^2 - 3x + 2

and

-2x^2 + 5x - 1

Solution

To subtract the polynomials, we need to change the sign of each term in the second polynomial. This means that we need to multiply each term by -1. So, the second polynomial becomes:

-2x^2 - 5x + 1

Now, we can subtract the terms:

x^2 - 3x + 2

-2x^2 - 5x + 1

Subtracting the terms, we get:

3x^2 + 2x - 3

Tips and Tricks

Here are a few tips and tricks to help you subtract polynomials:

Make sure to change the sign of each term in the second polynomial.
Subtract the terms by combining like terms.
Simplify the expression by combining like terms.
Check your work by plugging in values for the variables.

By following these tips and tricks, you can simplify complex expressions and arrive at a polynomial in standard form.

Common Mistakes

Here are a few common mistakes to avoid when subtracting polynomials:

Failing to change the sign of each term in the second polynomial.
Subtracting the terms incorrectly.
Failing to simplify the expression by combining like terms.
Not checking your work by plugging in values for the variables.

By avoiding these common mistakes, you can ensure that your answers are accurate and complete.