Subtracting Polynomials Worksheet, Steps, and Examples

Subtracting Polynomials: The Complete Guide

How to Subtract Polynomials

Subtracting polynomials involves taking one polynomial equation and subtracting another polynomial equation from it. The resulting polynomial equation is the difference between the two polynomials. Here are the steps to subtract polynomials:

1. Rewrite the polynomial equations in standard form. That is, arrange the terms in descending order of degree.
2. Change the signs of all the terms in the second polynomial equation. That is, replace all the plus signs with minus signs and vice versa.
3. Add the two polynomial equations together. This is done by adding the coefficients of the like terms. The resulting polynomial equation is the sum of the two polynomials.
4. Simplify the resulting polynomial equation by combining like terms.

It is important to follow the order of operations when subtracting polynomials. The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is:

1. Parentheses
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

When subtracting polynomials, it is important to remember that the order of operations applies to each polynomial equation individually. That is, you should simplify each polynomial equation before adding them together.

In summary, to subtract polynomials, you need to rewrite the polynomial equations in standard form, change the signs of all the terms in the second polynomial equation, add the two polynomial equations together, and simplify the resulting polynomial equation by combining like terms. Following the order of operations is crucial to ensure that the correct answer is obtained.

Steps in Subtracting Polynomials

Subtracting polynomials involves a few steps that need to be followed carefully in order to arrive at the correct answer. Here are the steps:

1. Arrange the polynomials vertically: The first step is to arrange the polynomials vertically, with the like terms lined up. This makes it easier to subtract the terms that are similar and cancel them out.

    

2. Change the signs of the second polynomial: The second polynomial needs to have all its signs changed. This is done by changing the sign of each term in the second polynomial.
3. Add the polynomials: The next step is to add the two polynomials together. This is done by adding each term of the first polynomial to its corresponding term in the second polynomial.
4. Combine like terms: The final step is to combine any like terms that result from adding the polynomials. This is done by adding the coefficients of the terms that have the same variables and exponents.

It is important to note that when subtracting polynomials, the order of the terms does not matter. The answer will be the same regardless of the order of the terms.

To evaluate the subtraction of polynomials, it is important to follow these steps carefully and double-check the answer to ensure accuracy. With practice, subtracting polynomials can become second nature and a valuable tool in solving algebraic equations.

Subtracting Polynomials Examples

Subtracting polynomials involves subtracting one polynomial from another. This is done by combining like terms and simplifying the resulting polynomial. Here are a few examples of subtracting polynomials:

Example 1

Subtract the polynomial (3x^2 + 2x – 5) from the polynomial (5x^2 – 3x + 7).

Solution

To subtract the polynomial (3x^2 + 2x – 5) from the polynomial (5x^2 – 3x + 7), we simply subtract the corresponding coefficients of the like terms. The resulting polynomial is:

(5x^2 – 3x + 7) – (3x^2 + 2x – 5) = 2x^2 – 5x + 12

Example 2

Subtract the polynomial (2x^3 – 5x^2 + x + 3) from the polynomial (4x^3 + 3x^2 – 2x – 5).

Solution

To subtract the polynomial (2x^3 – 5x^2 + x + 3) from the polynomial (4x^3 + 3x^2 – 2x – 5), we simply subtract the corresponding coefficients of the like terms. The resulting polynomial is:

(4x^3 + 3x^2 – 2x – 5) – (2x^3 – 5x^2 + x + 3) = 2x^3 + 8x^2 – 3x – 8

Example 3

Subtract the polynomial (x^4 + 2x^3 – 3x^2 + 5x – 7) from the polynomial (2x^4 – x^3 + 4x^2 – 2x + 3).

Solution

To subtract the polynomial (x^4 + 2x^3 – 3x^2 + 5x – 7) from the polynomial (2x^4 – x^3 + 4x^2 – 2x + 3), we simply subtract the corresponding coefficients of the like terms. The resulting polynomial is:

(2x^4 – x^3 + 4x^2 – 2x + 3) – (x^4 + 2x^3 – 3x^2 + 5x – 7) = x^4 – 3x^3 + 7x^2 – 7x + 10

With these examples, it’s clear that subtracting polynomials is a straightforward process of combining like terms and simplifying the resulting polynomial.

Subtracting Polynomials with Different Exponents

Subtracting polynomials with different exponents can be a bit tricky, but with some practice, it can become second nature. The key is to make sure that you are subtracting like terms. In other words, you want to make sure that the variables and their exponents are the same for each term.

To subtract polynomials with different exponents, you will need to follow these steps:

1. Rewrite the polynomials so that like terms are lined up. This means that you will need to rearrange the terms so that the variables and their exponents are in the same order for each term.

    

2. Change the sign of each term in the second polynomial. This means that if a term is positive, you will need to make it negative, and if a term is negative, you will need to make it positive.
3. Add the two polynomials together. This means that you will add the coefficients of each like term. If a term only appears in one polynomial, you can simply copy it over to the answer.

Let’s look at an example to see how this works:

(3x^2 + 2x – 5) – (4x^2 – 3x + 1)

First, we need to rearrange the terms so that like terms are lined up:

3x^2 + 2x – 5

-4x^2 + 3x – 1

Next, we need to change the sign of each term in the second polynomial:

3x^2 + 2x – 5

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

Finally, we can add the two polynomials together:

(3x^2 + (-4x^2)) + (2x + (-3x)) + (-5 + (-1))

-x^2 – x – 6

So the answer is -x^2 – x – 6.

It’s important to remember to always check your answer by simplifying it as much as possible. In this case, we can see that -x^2 – x – 6 cannot be simplified any further, so we know that our answer is correct.

Subtracting Polynomials with Exponents

Subtracting polynomials with exponents can be a bit trickier than adding them, but it’s still a straightforward process. The key is to remember that when you subtract one polynomial from another, you’re really just adding the opposite of the second polynomial.

Let’s take a look at an example:

(5x^3 + 2x^2 – 3x – 7) – (2x^3 – 4x^2 + 5x + 1)

To subtract these polynomials, you need to distribute the negative sign to each term in the second polynomial, like this:

5x^3 + 2x^2 – 3x – 7 – 2x^3 + 4x^2 – 5x – 1

Now you can combine like terms to simplify the expression:

(5x^3 – 2x^3) + (2x^2 + 4x^2) + (-3x – 5x) + (-7 – 1)

3x^3 + 6x^2 – 8x – 8

So the result of subtracting (2x^3 – 4x^2 + 5x + 1) from (5x^3 + 2x^2 – 3x – 7) is 3x^3 + 6x^2 – 8x – 8.

It’s important to remember that when subtracting polynomials, you should always double-check your work to make sure you haven’t made any mistakes. You can also use a table to help keep track of the terms and coefficients as you combine like terms.

In summary, subtracting polynomials with exponents involves distributing the negative sign to each term in the second polynomial and then combining like terms. Double-checking your work and using a table can help ensure accuracy.

Adding and Subtracting Polynomials

Adding and subtracting polynomials is a fundamental concept in algebra. A polynomial is a mathematical expression consisting of one or more terms, where each term is a product of a coefficient and one or more variables raised to a power. To add or subtract polynomials, you need to combine like terms.

Like Terms

Like terms are terms that have the same variables raised to the same powers. For example, 3x and -5x are like terms because they both have x raised to the first power. However, 3x and 4x^2 are not like terms because they have different powers of x.

Adding Polynomials

To add two or more polynomials, you simply add the coefficients of the like terms. For example, to add 2x^2 + 3x – 4 and x^2 – 2x + 5, you first group the like terms: (2x^2 + x^2) + (3x – 2x) + (-4 + 5). Then, you add the coefficients of the like terms: 3x^2 + x + 1. Therefore, the sum of 2x^2 + 3x – 4 and x^2 – 2x + 5 is 3x^2 + x + 1.

Subtracting Polynomials

To subtract one polynomial from another, you need to change the sign of all the terms in the second polynomial and then add the resulting polynomial to the first polynomial. For example, to subtract x^3 – 2x^2 + 3x – 4 from 2x^3 + 4x^2 – 5x + 6, you change the sign of all the terms in the second polynomial: -(x^3 – 2x^2 + 3x – 4) = -x^3 + 2x^2 – 3x + 4. Then, you add the resulting polynomial to the first polynomial: (2x^3 + 4x^2 – 5x + 6) + (-x^3 + 2x^2 – 3x + 4) = x^3 + 6x^2 – 8x + 10. Therefore, the difference between 2x^3 + 4x^2 – 5x + 6 and x^3 – 2x^2 + 3x – 4 is x^3 + 6x^2 – 8x + 10.

Simplifying Expressions

When adding or subtracting polynomials, it is important to simplify the resulting expression. This means combining like terms and reducing the expression to its simplest form. For example, to simplify the expression 4x^2 – 3x + 2x^2 + 5x – 7, you first group the like terms: (4x^2 + 2x^2) + (-3x + 5x) – 7. Then, you add the coefficients of the like terms: 6x^2 + 2x – 7. Therefore, the simplified form of 4x^2 – 3x + 2x^2 + 5x – 7 is 6x^2 + 2x – 7.

In conclusion, adding and subtracting polynomials involves combining like terms and simplifying the resulting expression. It is an important concept in algebra and is used in a variety of applications.

How to Subtract Polynomials FAQ

What is the difference between adding and subtracting polynomials?

The main difference between adding and subtracting polynomials is the operation used. When adding, you combine like terms by adding their coefficients. When subtracting, you combine like terms by subtracting their coefficients.

How do you subtract polynomials with like terms?

When subtracting polynomials with like terms, you simply subtract the coefficients of the like terms. For example, if you are subtracting 3x^2 – 2x^2, the result would be x^2.

Can you subtract polynomials with different exponents?

Yes, you can subtract polynomials with different exponents. However, you cannot subtract terms that are not like terms. For example, you cannot subtract 3x^2 from 5x.

Are there any special rules for subtracting polynomials?

There are no special rules for subtracting polynomials. However, it is important to remember to distribute the negative sign to all terms in the second polynomial before combining like terms.

What are some common mistakes to avoid when subtracting polynomials?

One common mistake to avoid when subtracting polynomials is forgetting to distribute the negative sign to all terms in the second polynomial. Another common mistake is not combining like terms correctly.

Is the order of subtraction important when subtracting polynomials?

No, the order of subtraction is not important when subtracting polynomials. However, it is important to remember to distribute the negative sign to all terms in the second polynomial before combining like terms.

What is the rule in subtracting polynomials?

The rule in subtracting polynomials is to distribute the negative sign to all terms in the second polynomial before combining like terms.

What is an example of subtracting polynomials?

An example of subtracting polynomials is (3x^2 + 4x – 5) – (2x^2 – 3x + 1) = x^2 + 7x – 6.