Multiplication and Division of Algebraic Expressions

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Introduction to Multiplication and Division of Algebraic Expressions

As the number of variables in an algebraic expression increase, and the higher powers of those variables begin to be used, the expressions begin to look more and more intimidating. \({2x + 5}\) looks simple enough but \({3x^3 + 4x^2y – 7xy^3 + y^3}\) seems quite scary, especially because there seems to be nothing that can be used to simplify it. After that, if you are asked to multiply or divide such algebraic expressions, the task might seem even more daunting. But the good news is that algebraic equations follow the same rules for multiplication and division as those which govern multiplication and division of numbers. Let us take a look at how this is done.

The Big Idea: What are multiplication and division of algebraic expressions?

We do know that an algebraic expression is a combination of coefficients, one or more variables (\({x, \,y, \,z}\) etc.), the powers to which those variables are raised (also called exponents), and one or more arithmetic operators (like \({\text{+, ‒, }x,\text{ or /}}\)). So it could be a monomial expression like \({3x}\), or it could be a binomial like \({4x + y}\), or it could have more terms and greater exponents, like \({3x^3 + 2xy\; – y^2 + 15}\).

Just like we need to add, subtract, multiply and divide numbers, we also need to carry out these same operations on algebraic equations as well. In this module, we will be discussing how to multiply and divide two algebraic expressions. They follow the same laws, rules and procedure. It just looks complicated, but in reality, it again is a test of patience and being methodical.

See, just like how you used to do it for numbers! It’s all about being systematic and patient.

Commutative Law

As you have already seen, multiplication of algebraic expressions follows the same pattern and flow of that of the multiplication of numbers, therefore, it makes sense that the laws that govern the operation there, apply here as well.

The commutative law of mathematics states that any finite sum or product is unaltered by reordering its terms or factors. What this means is that \({a + b}\) is the same as \({b + a}\). Similarly, \({a \times b}\) is the same as \({b \times a}\).

This same law is applicable in algebra also and applies to the multiplication of algebraic equations. That means

Associative Law

What happens when we need to multiply not two but three algebraic expressions together, in the format \({a \times b \times c}\)? There is a law that governs this kind of multiplication as well, and it is called the associative law. It states that the terms may be operated upon by associating them in any way desired. In other words, \({a(bc) = (ab)c}\).

Division of algebraic expressions

The process for dividing one polynomial by another polynomial is very similar to dividing one number by another. The most common method is the long division method.

While dividing polynomials, writing the dividend over the quotient in the traditional “fractional representation” isn’t very informative for all but the most basic expressions. For that reason, the long division approach is more universally applied. The steps are simple and vary ever so slightly from the long division of numbers. 

Have a look at the most systematic way to go about the long division of polynomials:

               

What constitutes the remainder and when do you know that you can’t divide a polynomial further? For that, think back to the definition of a polynomial. Only those algebraic expressions with positive powers of variables are termed as polynomials. So, whenever the terms remaining after step 6 is of an order smaller than the divisor, then it forms the remainder.

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