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

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.

How to multiply polynomials

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

Learn multiplication of polynomials

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}}

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

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

Multiplying monomials

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

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}

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

Multiply the coefficients and same variables separately

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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)}

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.

how to multiply two polynomials

Example 04
Multiply monomial and binomial

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

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

Binomial multiplication

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}}

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

Free learning of polynomial multiplication
\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}


how to multiply polynomial

\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)}


steps to multiply polynomial

Follow the below steps:

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

How to multiply binomials

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

Binomial multiplication for grade 6

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

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

Multiplication of polynomial for algebra

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)}


Free algebra lessons for kids

Follow the below steps

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

Methods to multiply polynomials

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

Polynomial algebra lessons for kids

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

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

Polynomial multiplication example

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

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