12.3. Division of Algebraic Expressions

We have learnt how to add and subtract algebraic expressions. We also know how to multiply two expressions. We have not however, looked at division of one algebraic expression by another. This is what we wish to do in this section.

In this section, we will explore how to divide algebraic expressions, which is an important operation alongside addition, subtraction, and multiplication. Division is basically the reverse of multiplication, and understanding this relationship helps when working with algebraic expressions. Just as you might divide numbers, you can divide expressions as well.

Think of division like sharing a pizza. If you have a whole pizza (an algebraic expression) and you want to share it among a certain number of people (like dividing by another expression), you'd be looking at how much each person gets, similar to how we find a quotient in division.

Now we shall look closely at how the division of one expression by another can be carried out. To begin with we shall consider the division of a monomial by another monomial.

When dividing one monomial by another, we can break down each monomial into its factors and simplify. For instance, in the expression 6x³ ÷ 2x, we can factor both the numerator and the denominator and cancel out the common factors. This leads to a simpler expression as a result.

Imagine you have 6 apples and you want to give them to 2 friends. If you divide the apples evenly, each friend gets 3 apples. This is similar to how we cancel out factors in an algebraic expression!

Consider 6x³ ÷ 2x. We may write 2x and 6x³ in irreducible factor forms, 2x = 2 × x, 6x³ = 2 × 3 × x × x × x. Now we group factors of 6x³ to separate 2x, 6x³ = 2 × x × (3 × x × x) = (2x) × (3x²).

Using the example of 6x³ ÷ 2x, we identified that both expressions can be written in terms of their prime factors. This allows us to easily see that 2x is a common factor, which we can cancel out. This leads us to the final result of 3x². The process of simplifying these expressions using their factors is key in dividing algebraic expressions.

Think of this process like sorting blocks. If you have 6 different blocks (x-factors) in a pile but you can group them with a smaller set (2x), it's easier to see what remains after taking out the common ones!

Let us consider the division of the trinomial 4y³ + 5y² + 6y by the monomial 2y. Therefore, (4y³ + 5y² + 6y) ÷ 2y.

When dividing a polynomial by a monomial, we can either factor out the common divisor from each term of the polynomial or divide each term separately by the monomial. This allows us to simplify each term and combine them back into a polynomial, which is often much easier to work with.

Consider this as distributing candies. If you have a bag of candies (the polynomial) and you want to share them (divide by the monomial), you would count how many candies each friend receives and see what’s left over by dividing each count individually.

In the case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the divisor polynomial. Instead, we factorise both the polynomials and cancel their common factors.

When you divide one polynomial by another polynomial, it's not just as simple as dividing term by term. The correct approach is to factor each polynomial first. This helps identify any common factors that can be cancelled, simplifying the problem much like reducing fractions.

Imagine you have a recipe that makes several servings (the polynomial), and you want to share this recipe with a friend (dividing by another polynomial). By breaking down the ingredients (factoring), you can see what parts you both can share or keep to make perfect servings.

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