Introduction
Instructions
The section consists of different components:
(a) For problems: Enter the answers in the blanks provided and use the forward button to proceed further with the respective question.
(b) Equation blanks: Upon clicking on the blank, you will be provided with a variety of algebraic operations. Use these operations to enter the appropriate answer. For example: To enter perimeter of rectangle, 2(l + b), we need to put '2 x (l+b)'. Make sure to use the bracket to be clear about the entities that are getting operated on.
In the earlier class we have discussed the concepts of:
Factors Coefficients Monomials Binomials Trinomials
We have also tried to find the value for the expressions using trial and error method. The expressions that we dealt with were much simpler than the ones we usually end up encountering when doing analysis in real life.
Complex phenomenon need likewise mathematical expressions for analysis and understanding.
When dealing with complicated equations, it is better to simplify those whilst working with them. We will be looking at one of the many different methods to simplify algebraic expressions (or polynomials) in this chapter. That methiod being factorisation. Re-writing the expression as a product of its factors is known as Factorisation.
Factors of natural numbers
Let's review some of the concepts that we have learnt earlier and later segue into the factorisation.
Taking a natural number, for eg- 30, we can now express it as a product of its natural number factors like:
30 =
Thus, 1, 2, 3, 5, 6, 10, 15 and 30 are the factors of 30.
Out of these, 2, 3 and 5 are the prime factors of 30 because
A number written as a product of prime factors is said to be in the prime factor form.
Trying to now express 30 as a product of its prime factors we can write:
30 =
Similarly the prime factorisation of 70 and 90:
70 =
90 =
Like we express numerical values as products of their prime factors- the same way, we can express algebraic expressions as products of their factors. We will be learning to do in this chapter.
Factors of algebraic expressions
We are aware that in algebraic expressions, terms are a product of its factors. For example: in 5xy + 3x
5xy =
Is there an important observation here? We can see that the factors 5, x and y of 5xy cannot be further be expressed as products of smaller factors. In other words, we can say that 5, x and y are ‘prime’ factors of 5xy.
In the case of algebraic expressions, we use the word ‘irreducible’ in place of ‘prime’. Thus,
5 × x × y is the irreducible form of 5xy.
Is 5 × (xy) an irreducible form of 5xy ?
5 × (xy) is not an irreducible form of 5xy as the factor 'xy' can be further expressed as a product of x and y i.e. xy = x × y.
Note: 1 is a factor of any number.
Therefore, when we write a number as a product of factors, we do not write 1 as a factor (unless it is required).
Similarly, for say, '5xy'- 1 is again a factor. Even here, we do not show 1 as a separate factor.
Consider the expression 3x(x + 2). It can be written as a product of factors :
3x(x + 2) =
The factors 3, x and (x +2) are irreducible factors of the expression- 3x (x + 2).
Similarly, for the expression- 10x (x + 2) (y + 3) can be expressed in its irreducible factor form as:
10x (x + 2) (y + 3) =