How to Add and Subtract Polynomials Step by Step. Resolved exercises.

In this post we will learn how to add and subtract polynomials step by step and we will analyze the small difficulties that can be presented and how to solve them.

Next I will explain the addition and subtraction of polynomials, with examples resolved step by step.

But first, what are the terms of a polynomial?

What are the Terms of a Polynomial

Before we begin to explain how to add and subtract polynomials we must be very clear about what the terms of a polynomial are.

A term is composed of:

  • Coefficient: A number (if none appears, it has a 1)

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  • Literal part: One or more variables, which may be elevated to some exponent or not.

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We differentiate them because they are separated by the signs of addition and subtraction in the polynomial.

A polynomial of a single term is called a monomial, with two terms it is called a binomial and with three terms trinomial.

Similar Terms

Similar terms are those that have exactly the same literal part, i. e. the same variables elevated to the same exponents. If the slightest difference existed, it would no longer be similar terms.

For example, the terms of each line are similar to each other:

polinomios ejemplos

Have you seen? The literal part is exactly the same (although in the second line they are not in the same order. What matters is that they have the same variables elevated to the same exponents).

However, these two terms are not alike:

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They seem to be, but they are not, since the variables m and n are not elevated to the same exponent.

How to add and subtract monomials

When you learn how to add and subtract numbers, they teach you to add apples with apples and oranges with oranges.

To add and subtract monomials, something similar happens, since you must bear in mind that only similar terms can be added, that is to say, as we have indicated before, the terms with the same literal part. That is why it is so important to master the concept of such a term. To be able to differentiate when it is possible to make the addition or subtraction.

Really, when we add and subtract similar terms, we are adding or subtracting the coefficients

Then, in order to add or subtract similar terms, the following steps are followed:

1. Make sure that the terms are similar

2. Add or subtract the coefficients before each similar term

3. Keep the literal part

Let’s solve a simple step-by-step example to help you better understand how it works:

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In this case, we have two terms: and 3a².

In the termwe have:

  • Coefficient: 1 (when it has nothing is that it has a 1)
  • Literal part:

In the term 3a² we have:

  • Coefficient: 3
  • Literal part:

In both terms, the literal part is a² and is exactly the same. Therefore, they are similar terms and can be added together, so:

2. We add the coefficients: 1 + 3 = 4
3. We keep the literal part:

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These steps, when you have more practice are done by heart, but to begin and understand them well, I indicate it in detail.

Let’s look at another example very similar to the previous one, but with a small difference:

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We have two terms: and 3a³

In the term we have:

  • Coefficient: 1
  • Literal part:

En el término 3a³ tenemos:

  • Coefficient: 3
  • Literal part: 3a³

Now the literal part of both terms are not exactly the same. The variables coincide, but are not elevated to the same exponent, since one is squared and the other to the cube.

Therefore, these terms are not similar and cannot be added together.

How to add and subtract polynomials

There are polynomials in which we have several terms with different literal parts, such as for example:

suma y resta de polinomios ejemplos

In these cases, we must identify whether there are similar terms between them and add or subtract them separately.

Each of these additions and subtractions of similar terms will form the final polynomial terms.

Let’s take it slower:

In the previous polynomial there are three types of terms:

  • Terms with literal part x
  • Terms with  literal part y
  • Terms without literal part

Terms with literal part x

We have two terms with the literal part x:

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Having the same literal part, they are similar terms and can therefore be added or subtracted from each other. Then:

  • Subtract the coefficients: 2-1=1
  • We keep the literal part: x

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The resulting term is one of the terms of the final result.

Terms with literal part y

We have two terms with the literal part y:

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Having the same literal part, they are similar terms and can therefore be added or subtracted from each other. Then:

  • We subtract the coefficients: -3+1=-2
  • We keep the literal part: y

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The resulting term is another of the final polynomial terms.

Terms without literal part

Terms that have no literal part are:

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Terms without literal part are numbers, which we operate with them and the result will be the last term of the final polynomial:

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And in this way we have obtained the final result, adding or subtracting all similar terms from each other, as I indicated at the beginning.

Separate addition and subtraction of polynomials

There are times when they ask us to add and subtract polynomials, but they give us each of them separately.

For example: Add and subtract the following polynomials:

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Sum of polynomials: Example solved

Let’s start by explaining how to add polynomials:

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First, we replace P (x) and Q (x) with their terms:

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We eliminate the parentheses, taking into account the rule of the signs.

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Now, we add and subtract the similar terms from each other and it remains:

ejemplos de sumas de polinomios

Polynomial Subtraction: Resolved Example

Once we’ve seen how polynomials add up, let’s see how to subtract polynomials:

As before, we replaced P (x) and Q (x) with their terms:

And we eliminated parentheses. In this case, we have a minus sign in front of one of them, which modifies the sign of the terms inside the parentheses, since it is equivalent to multiplying by -1:

Once we have the polynomial without parentheses, we can add and subtract the similar terms from each other:

And that’s how easy it is to add and subtract polynomials.

Resolved exercises of addition and subtraction of polynomials

1 – Performs the following polynomial operations:

2 – Given the polynomials:

Perform:

 

Exercise 1:

Exercise 2:

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