Factorisation is the process of breaking down algebraic expressions into simpler expressions or factors. These factors can be multiplied together to give back the original expression. This chapter covers different techniques for factorising polynomials using common factors, regrouping terms, identities, and quadratic expressions.
Factorisation by Common Factors
The easiest method to begin factorising is by extracting common factors from the terms in an algebraic expression.
Example
Problem: Factorise 15xy + 20x2y2.
Process:
Identify the common factors in the terms. The greatest common factor here is 5xy.
Factorise: 15xy + 20x2y2 = 5xy(3 + 4xy).
Answer: 5xy(3 + 4xy)
Factorisation Using Identities
Factorisation can often be simplified using standard algebraic identities. Here are three commonly used identities:
a2 – b2 = (a – b)(a + b)
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Example
Problem: Factorise 25x2 – 16y2.
Process:
Recognise that the expression fits the identity a2 – b2 = (a – b)(a + b).
When an expression does not directly fit an identity or involve common factors, regrouping terms can help.
Example
Problem: Factorise x3 – 3x2 + 2x – 6.
Process:
Group the terms: (x3 – 3x2) + (2x – 6).
Factorise each group: x2(x – 3) + 2(x – 3).
Factor out the common binomial: (x2 + 2)(x – 3).
Answer: (x2 + 2)(x – 3)
Factorisation Using the Splitting the Middle Term Method
This method is primarily used for quadratic polynomials and involves splitting the middle term of a quadratic equation into two terms that help in factorisation.
Example
Problem: Factorise 2x2 + 5x + 3.
Process:
Multiply the coefficient of x2 (2) and the constant term (3). The product is 6.
Find two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3.
Word problems involving factorisation often require translating a given situation into an algebraic expression and then factorising it.
Example
Problem: The product of two consecutive even numbers is represented by x(x + 2). If their product is 48, find the numbers.
Process:
Set up the equation: x(x + 2) = 48.
Simplify: x2 + 2x – 48 = 0.
Use the quadratic formula or factorisation: (x – 6)(x + 8) = 0.
The solutions are x = 6 and x = -8. Therefore, the consecutive even numbers are 6 and 8.
Answer: The consecutive even numbers are 6 and 8.
Important Mathematical Problems
Problem 1
Factorise 6x2 – 7x – 3.
Problem 2
Factorise 5x2 + 13x + 6 using the splitting middle term method.
Key Terms and Concepts
Common Factor: A term that divides all the terms in an expression.
Quadratic Expression: A polynomial of degree 2.
Algebraic Identity: A standard formula used to simplify expressions.
Factorisation: Breaking down an expression into simpler factors.
FAQs on Chapter 12: Factorisation
1. What is factorisation?
Factorisation is the process of breaking down an algebraic expression into simpler factors that, when multiplied together, give the original expression.
2. What are common factors in factorisation?
Common factors are the terms or expressions that are shared by two or more terms in an algebraic expression and can be factored out.
3. What is the identity for factorising a2 – b2?
The identity for factorising a2 – b2 is (a – b)(a + b).
4. How do you factorise using the common factor method?
To factorise using common factors, find the greatest common factor (GCF) of all the terms in the expression, and then divide each term by the GCF, placing it outside the bracket.
5. How do you factorise quadratic expressions?
Quadratic expressions are often factorised by recognising them as a product of two binomials, using identities or by splitting the middle term.
6. What is regrouping in factorisation?
Regrouping is a method of factorising where the terms of an expression are grouped in pairs or sets to simplify factorisation.
7. What is the identity for (a + b)2?
The identity for (a + b)2 is a2 + 2ab + b2.
8. How do you factorise x2 + 7x + 12?
Find two numbers whose sum is 7 and product is 12, which are 3 and 4. Therefore, x2 + 7x + 12 = (x + 3)(x + 4).
9. What is the identity for (a – b)2?
The identity for (a – b)2 is a2 – 2ab + b2.
10. What is the splitting the middle term method?
The splitting the middle term method involves splitting the middle term of a quadratic expression into two parts that add up to the middle term and help in factorisation.
11. How do you factorise 25x2 – 16y2?
Recognise the expression as a2 – b2 and apply the identity: 25x2 – 16y2 = (5x – 4y)(5x + 4y).
12. What are algebraic identities?
Algebraic identities are standard formulas used to simplify expressions, such as (a + b)2 = a2 + 2ab + b2 and a2 – b2 = (a – b)(a + b).
13. What is the first step in factorising an expression?
The first step in factorising an expression is to check for any common factors that can be factored out from all the terms.
14. What is a quadratic expression?
A quadratic expression is a polynomial of degree 2, usually written in the form ax2 + bx + c, where a, b, and c are constants.
15. How do you factorise expressions with four terms?
Expressions with four terms can be factorised by grouping the terms into pairs and then factoring out the common factors from each group.
16. How do you solve word problems using factorisation?
To solve word problems using factorisation, translate the problem into an algebraic expression and then apply factorisation techniques to simplify and solve.
17. Can you factorise expressions without common factors?
Yes, even if there are no common factors, expressions can still be factorised using techniques like regrouping, identities, or the splitting the middle term method.
18. What are consecutive even numbers?
Consecutive even numbers are even numbers that follow each other, such as 2, 4, 6, or 6, 8, 10. Their difference is always 2.
19. How do you factorise x(x + 2) = 48?
Expand the equation to get x2 + 2x – 48 = 0 and then factorise: (x – 6)(x + 8) = 0. The solutions are x = 6 and x = -8.
20. What is the greatest common factor?
The greatest common factor (GCF) is the largest factor that divides all the terms in an expression.
21. How do you factorise using the identity a2 – b2?
To factorise using the identity a2 – b2, apply the formula: a2 – b2 = (a – b)(a + b).
22. How do you factorise 6x2 – 7x – 3?
Multiply the coefficient of x2 by the constant (-3) to get -18. Find two numbers that multiply to -18 and add to -7. These are -9 and 2. Factorise: 6x2 – 9x + 2x – 3 = 3x(2x – 3) + 1(2x – 3) = (3x + 1)(2x – 3).
23. Can you apply factorisation to real-life problems?
Yes, factorisation is used in real-life situations like solving problems related to area, speed, or finance, where expressions need to be simplified or equations need to be solved.
24. What is the difference between factorisation and expanding?
Factorisation is breaking down an expression into factors, while expanding is multiplying factors to obtain the original expression.
25. What is the importance of factorisation in mathematics?
Factorisation is important in mathematics as it simplifies complex algebraic expressions, making it easier to solve equations, understand relationships, and apply them to real-life problems.
MCQs on Chapter 12: Factorisation
MCQs on Chapter 12: Factorisation
1. What is factorisation?
2. What is the identity used to factorise a2 – b2?
3. Which of the following is a common factor of 15x and 25x?
4. What is the factorisation of 25x2 – 16?
5. How do you factorise x2 + 7x + 12?
6. Factorise x2 – 4x + 4
7. Which identity is used to factorise x2 – 9?
8. How do you factorise 6x2 + 7x – 3?
9. What is the factorisation of x2 + 9x + 14?
10. What is a common factor of 12x2 and 18x?
11. Factorise 9x2 – 16
12. Which of the following is a factor of 5x + 15?
13. How do you factorise x2 + 5x – 24?
14. Factorise 2x2 – 8x
15. Factorise 4x2 – 9y2
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