Multiplication of Algebraic expressions

A mathematical operation of multiplying two or more algebraic expressions is called the multiplication of algebraic expressions.

Introduction

In algebra, two or more algebraic expressions are involved in multiplication to represent their product. The multiplication of them is expressed in mathematical form by displaying a multiplication sign ($\times$) between every two expressions. The product of any two algebraic expressions is calculated by multiplying an expression by another and this process is repeated if more than two expressions are involved in the multiplication.

Required knowledge

For multiplying the algebraic expressions, it is essential to have a knowledge on the following concepts.

Procedure

There are three steps involved in finding the product of any two multiplying algebraic expressions.

1. Distribute the multiplication of an algebraic expression to each term of another algebraic expression.
2. Distribute the multiplication over addition or subtraction to find the products of the terms.
3. Check the expression for the like terms. if it contains like terms, then combine the terms.

Repeat the same process to find the product of more than two multiplying algebraic expressions.

Example

$ax+by$ and $cx+dy+e$ are two algebraic expressions.

First step

Now, display a multiplication sign between them to express the multiplication of them in mathematical form.

$(ax+by) \times (cx+dy+e)$

Take $m = cx+dy+e$

$=\,\,\,$ $(ax+by) \times m$

$=\,\,\,$ $m \times (ax+by)$

Now, use distributive property to distribute the multiplication.

$=\,\,\,$ $m \times ax + m \times by$

$=\,\,\,$ $ax \times m + by \times m$

$=\,\,\,$ $ax \times (cx+dy+e)$ $+$ $by \times (cx+dy+e)$

$\therefore \,\,\,$ $(ax+by) \times (cx+dy+e)$ $\,=\,$ $ax \times (cx+dy+e)$ $+$ $by \times (cx+dy+e)$

Therefore, this equation has cleared that the first step can be directly obtained by multiplying each term of one expression with another expression.

Second step

$=\,\,\,$ $ax \times (cx+dy+e)$ $+$ $by \times (cx+dy+e)$

Now, use distribution property one more time.

$= \,\,\,$ $ax \times cx$ $+$ $ax \times dy$ $+$ $ax \times e$ $+$ $by \times cx$ $+$ $by \times dy$ $+$ $by \times e$

$= \,\,\,$ $acx^2$ $+$ $axdy$ $+$ $axe$ $+$ $bycx$ $+$ $bdy^2$ $+$ $bye$

$= \,\,\,$ $acx^2$ $+$ $adxy$ $+$ $aex$ $+$ $bcxy$ $+$ $bdy^2$ $+$ $bey$

$= \,\,\,$ $acx^2$ $+$ $bdy^2$ $+$ $aex$ $+$ $bey$ $+$ $adxy$ $+$ $bcxy$

The product of the algebraic expressions is also an algebraic expression.

Third step

Lastly, simplify the algebraic expression by the fundamental operations of terms for combining the like terms.

$= \,\,\,$ $acx^2$ $+$ $bdy^2$ $+$ $aex$ $+$ $bey$ $+$ $(ad+bc)xy$

Thus, the algebraic expressions are multiplied mathematically to obtain their product in algebra.

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