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?

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**

The below points will help you multiply two algebraic entities.

(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 01Multiplication 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 02Multiplication 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 04Multiply 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

**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**

Follow the below steps:**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}

Adding the like terms.

\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**

Follow the below steps

**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.