Now, I’m going to explain to you **how to multiply polynomials** step by step.

I’ll explain to you the multiplication of polynomials from the beginning, from the most basic to the most complicated.

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## How to Multiply a Number by a Monomial

This is the simplest case. We proceed in the same way as for multiplying two terms, but in this case, one of the two terms is only a number.

Therefore, the numbers are multiplied and the monomial variables remain as they are in the monomial.

For example:

1. Multiply the numbers:

2. The variables and exponents of the monomial remain the same:

And we have the final result.

## How to Multiply a Monomial by a Monomial

We recall that a monomial is a single-term polynomial.

Therefore, multiplying two monomials is the same as multiplying two terms and proceeds in the same way as explained above to multiply two terms.

As you know, the terms of a polynomial are composed of a coefficient (number and sign) and the literal part (variables elevated to their exponents).

To multiply two terms, follow this procedure:

1. The coefficients of each term are multiplied, as if they were multiplying integers, taking into account the rule of signs.

2. To multiply the literal part of each term, it is necessary to treat them as a multiplication of powers with different bases: for variables that are equal, the base is maintained and the exponents are added and those that are not equal remain as they are in the result. Let’s take a slower look at the procedure with an example:

In this power multiplication we have two bases x e and:

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Now for powers having the same base, the exponents are added: x on the one hand e and on the other hand:

Let’s see another example of how to multiply the terms of a polynomial, with terms that also have a coefficient:

First, we multiply the coefficients, just as we multiply the integers:

Now we go to the literal part. In this case, we have 3 bases: a, b and c. (Although the variables do not appear elevated to any exponent, their exponent is actually 1). Therefore, for the bases that are equal, their exponents are added, as in the previous example:

## How to Multiply a Number by a Polynomial

To multiply a number by a polynomial, multiply the number by each of the polynomial terms.

Let’s see an example step by step:

1. The number is multiplied by the first term of the polynomial:

2. The number is multiplied by the second term of the polynomial:

3. The number is multiplied by the third term of the polynomial:

If we had more terms, it would be necessary to follow this way successively.

### Less Sign in Front of a Polynomial in Parentheses

A particular case of a number by a polynomial is that of a minus sign in front of a polynomial in parentheses.

It is equivalent to multiplying by -1:

As you can see, the minus sign changes the terms of the original polynomial. This is a good way to remember to remove parentheses when operating.

## How to Multiply a Monomial by a Polynomial

In this case, we have to multiply the monomial by each of the terms of the polynomial, as we have done before. That is to say, it is to go multiplying monomial by monomial repeated times.

For example:

1. The monomial is multiplied by the first term of the polynomial:

2. The monomial is multiplied by the second term of the polynomial:

3. The monomial is multiplied by the third term of the polynomial:

## Multiplication of Polynomials. Polynomial by Polynomial

We have already reached the most complicated case to learn how to multiply polynomials. Here you have to multiply each term of one polynomial by all the terms of the other polynomial, i.e.:

- The first term of a polynomial is multiplied by all the terms of the other polynomial
- The second term of one polynomial is multiplied by all the terms of the other polynomial
- The third term of one polynomial is multiplied by all the terms of the other polynomial
- Y so on…
- Terms are regrouped, adding and subtracting similar terms

Let’s see some examples:

1. We multiply the first term of the first polynomial by all the terms of the second polynomial:

2. Now, we multiply the second term of the first polynomial by all the terms of the second polynomial:

3. We group similar terms together:

When we multiply two equal binomials (which is the square of a binomial), instead of developing them through multiplication, the formulas of notable products are used to save time and arrive at the result more directly. For example:

(a+b)(a+b) = (a+b)² = a² + 2ab + b²

Finally, let’s see the example of multiplying two polynomials with three terms each:

1. We multiply the first term of the first polynomial by all the terms of the other polynomial:

2. We multiply the second term of the first polynomial by all the terms of the other polynomial:

3. We multiply the third term of the first polynomial by all the terms of the other polynomial:

4. We group similar terms

Observa como el término con x elevada a 4 a desaparecido, porque los dos términos que aparecen eran iguales pero de signo contrario, por tanto seulan.