12.1.2. Factors of algebraic expressions

We have seen in Class VII that in algebraic expressions, terms are formed as products of factors. For example, in the algebraic expression 5xy + 3x the term 5xy has been formed by the factors 5, x and y, i.e., 5xy = 5 × x × y.

In algebra, expressions like 5xy + 3x consist of terms that can be broken down into factors. Just as 30 can be expressed as products of natural numbers, such as 2 × 15 or 3 × 10, algebraic terms can also be expressed as products of their constituent factors. For the term 5xy, the factors include the coefficient (5) and the variables (x and y). This means you can think of 5xy as being built from the parts 5, x, and y, just like you could break down a number into smaller numbers that multiply together to give you the original number.

Imagine you have a toy that is made up of three different parts: a wheel (5), a left wing (x), and a right wing (y). Just as the toy can't function without any one of these pieces, the term 5xy can't be represented without its factors. So, if you wanted to talk about this toy to your friend, you might say it’s a wheel attached to a left and right wing, just like saying 5xy is made from 5, x, and y.

In fact, 1 is a factor of every term. As a special case, we use the term ‘irreducible’ instead of ‘prime’ for factors in algebraic expressions. We say that 5 × x × y is the irreducible form of 5xy.

In algebraic expressions, we call factors that cannot be broken down any further as 'irreducible'. While in number theory we often discuss 'prime factors' (numbers that can only be divided by 1 and themselves), for algebraic expressions, we use 'irreducible' to emphasize that the factors are 'built' and cannot be simplified further. For example, in the expression 5xy, the factors 5, x, and y are considered irreducible, as there are no further divisions or simpler forms of these components.

Consider a Lego set where a car is built with a wheel, a body, and a steering wheel. Each of these parts is essential and cannot be further broken down without losing its functionality. Just like the car's irreducible parts, the terms in an algebraic expression like 5xy are essential for its structure.

Next consider the expression 3x (x + 2). It can be written as a product of factors: 3, x and (x + 2). The factors 3, x and (x + 2) are irreducible factors of 3x (x + 2).

In this example, the expression 3x(x + 2) can be broken down into its basic components: 3, x, and the polynomial (x + 2). Just like how we previously discussed the parts of 5xy, here each part (3, x, and (x + 2)) plays a crucial role in forming the full expression. Since neither 3, x, nor (x + 2) can be simplified any further, they are all considered irreducible.

Think of this like making a smoothie with three main ingredients: an apple (3), a banana (x), and a blend of spinach and kale (x + 2). Each ingredient contributes to the flavor of the smoothie, and you can't simplify any of them further without losing the essence of the smoothie. Thus, they are like the irreducible factors in our algebraic expression.

Similarly, the expression 10x (x + 2) (y + 3) is expressed in its irreducible factor form as 10x (x + 2) (y + 3) = 2 × 5 × x × (x + 2) × (y + 3).

When we take the expression 10x(x + 2)(y + 3), we can break down 10 into its prime factors, which are 2 and 5. This results in a full factorization where each component can stand alone without any further breakdown. The fact that we can express 10 as a product of irreducible factors complements our understanding of how to approach similar expressions.

Imagine building a garden with 10 individual plants, where each plant has a base made of soil (2), a trunk made of wood (5), and leaves (added variables). Each aspect of your garden represents a part of the whole; without any single aspect, the structure of your garden would be fundamentally changed, just like how each factor contributes equally to the expression.

--