# Multiplying polynomials || How to multiply polynomials

In this post we will learn methods to multiply two or more polynomials.

This is very important chapter in algebra.

To understand the post you should have basic understanding of algebraic expressions, constants, variables and coefficients.

## Methods of Multiplying Polynomials

Before understanding the multiplication, let us first revise the concept of polynomials.

### What are polynomials?

The collection of one or more algebraic entity separated by addition/subtraction is known as polynomial.

The entity can be constant, variables or mixture of both.

#### What are types of polynomial?

Different types of polynomial are:

(a) Monomial
The polynomial containing one entity is called monomial.

(b) Binomial
The polynomial with two entity is called Binomial

(c) Trinomial
The polynomial with three entity is called trinomial.

And the number goes on and on.

### Tools to multiply polynomials

(a) Multiplication of two entities is done by multiplying the coefficients and same variables separately.

(b) Multiplication of same variables is done by rules of exponents.

Some of the important exponent rules are:

(i) The multiplication of exponents with same base can be done by adding the powers.

(ii) Any exponent raised to the second power is simplified by multiplying the two powers.

Let us now move to solve some examples.
The below examples will prepare you to solve the questions on your own

### Examples of Multiplying Polynomials

Example 01
Multiplication of Monomials.

Multiply \mathtt{3x\ \&\ 5x^{3}}

Solution
\mathtt{\Longrightarrow \ 3x\ \times \ \ 5x^{3}}

To multiply the given entities, separate the coefficients and same variables and then multiply.

Hence, \mathtt{15x^{4}} is the solution.

Note:
Use the above mentioned law of exponent to multiply variables with the same base.

Example 02
Multiplication of Monomials

Multiply \mathtt{\ 9a^{3} bc^{2} \ \&\ \ 2a^{4} c}

Solution
\mathtt{9a^{3} bc^{2} \ \times \ \ 2a^{4} c}

Multiply the coefficients and same variables separately

Note that:
Variable \mathtt{a^{3}} is multiplied with \mathtt{a^{4}} since both of them have same base.

Similar is the case with \mathtt{c^{2} \ \&\ c}

Example 03
Multiplying Monomial and Binomial.
Multiply \mathtt{10a^{2} \ \&\ \ \left( 3a^{2} b\ +\ 2a\right)}

Solution
we know that:
Monomial contains one entity.
Binomial contains two entity separated by addition/subtraction.

Here the entity of monomial will be multiplied with both the entities of binomial.

Example 04
Multiply monomial and binomial

\mathtt{-3y^{3} z\ \&\ \ \left( x^{2} y\ -\ 5yz^{2}\right)}

Solution
The single entity of the monomial will be multiplied by both entity of binomial.

Hence, \mathtt{-3x^{2} y^{4} z\ \ +\ 15y^{4} z^{3}} is the solution.

#### Binomial Multiplication – Direct Method

Example 05
Multiply binomial with binomial
\mathtt{x^{2} +2xy\ \ \&\ \ xy^{3} +8xy^{2}}

Solution
Both the given terms are binomials.

To multiply the above binomials, follow the below step:

Consider that the first binomial is made of two monomial.

(a) Multiply the first entity of first binomial with second binomial.
(b) Then multiply second entity of first binomial with the second binomial

\mathtt{=x^{2}\left( xy^{3} +8xy^{2}\right) \ +\ 2xy\left( xy^{3} +8xy^{2}\right)}\\\ \\ \mathtt{=\left( x^{2} .xy^{3} +x^{2} .8xy^{2}\right) \ +\ \left( 2xy.xy^{3} +2xy.8xy^{2}\right)}\\\ \\ \mathtt{=\ \left( x^{3} y^{3} +8x^{3} y^{2} \ \right) +\ \left( 2x^{2} y^{4} +16x^{2} y^{3}\right)}\\\ \\ \mathtt{=x^{3} y^{3} +8x^{3} y^{2} \ +\ 2x^{2} y^{4} +16x^{2} y^{3}}

Example 06
Multiply binomial with binomial
\mathtt{\ 9\ -\ y\ \ \&\ \ x^{2} y+2xy}

Solution

\mathtt{=9\left( x^{2} y+2xy\right) \ -y\ \left( x^{2} y+2xy\right)}\\\ \\ \mathtt{=\left( 9x^{2} y+18xy\right) \ -x^{2} y^{1+1} -2xy^{1+1}}\\\ \\ \mathtt{=9x^{2} y+18xy\ -x^{2} y^{2} -2xy^{2}}

#### Binomial Multiplication – Detailed Solution

Example 07
Multiply the given binomials.
\mathtt{( x\ +\ 2) \ \ \&\ \ ( x+3)}

Solution

Step 01
Multiply 1st term of 1st binomial with 1st term of second binomial.

Step 02
Multiply 1st term of 1st binomial with 2nd term of 2nd binomial

Step 03
Multiply 2nd term of 1st binomial with 1st term of 2nd binomial.

Step 04
Multiply 2nd term of 1st binomial with 2nd term of 2nd binomial

Hence on multiplying the two given binomials, we get following results;

\mathtt{\Longrightarrow \ x^{2} +\ 3x\ +\ 2x\ +\ 6}

\mathtt{\Longrightarrow \ x^{2} +\ 5x\ +\ 6}

Hence, the above expression is the solution.

Example 08
Multiply the given binomials
\mathtt{\left( xy-3x^{3} y\right) \ \left( 9xy\ -2xy^{3}\right)}

Solution

Step 01
Multiply 1st term of 1st binomial with 1st term of second binomial.

Step 02
Multiply 1st term of 1st binomial and 2nd term of 2nd binomial

Step 03
Multiply 2nd term of 1st binomial and 1st term of second binomial

Step 04
Multiply 2nd term of 1st binomial with 2nd term of 2nd binomial

Hence, \mathtt{=9x^{2} y^{2} -2x^{2} y^{4} \ -27x^{4} y^{2} +6x^{4} y^{4}} is the solution.