FACTORISATION OF ALGEBRAIC EXPRESSIONS

Introduction

Factorisation is the reverse process of multiplication (expansion). It involves writing an algebraic expression as a product of its factors. This is a crucial skill for solving equations and simplifying rational expressions.

Brief Overview

graph TD; Factorisation --> Common[Common Factors]; Factorisation --> Grouping[By Grouping]; Factorisation --> Identities[Using Identities]; Factorisation --> Quadratics[Quadratic Trinomials];

1. Common Factors

Method: Find the Greatest Common Factor (GCF) of all the terms in the expression and "pull it out" using the distributive property in reverse: $ab + ac = a(b + c)$.

Numerical Example:
Factorise $6x^2y + 9xy^2$.
The GCF of 6 and 9 is 3. The GCF of $x^2y$ and $xy^2$ is $xy$.
So, GCF = $3xy$.
Pulling it out: $3xy(2x + 3y)$.

MCQs - Common Factors

What is the greatest common factor of $8x^3$ and $12x^2$?
a) $4x$
b) $4x^2$
c) $2x^3$
d) $24x^2$
Answer: b
Factorise $5a - 15$.
a) $5(a - 3)$
b) $5(a - 15)$
c) $3(a - 5)$
d) $5a(1 - 3)$
Answer: a
Pull out the common factor from $x^3 + x^2 + x$.
a) $x(x^2 + x)$
b) $x^2(x + 1 + 1/x)$
c) $x(x^2 + x + 1)$
d) $x(x^2 + x + 0)$
Answer: c
Factorise $14a^2b - 7ab^2$.
a) $7ab(2a - b)$
b) $7a(2ab - b^2)$
c) $7b(2a^2 - ab)$
d) $14ab(a - b)$
Answer: a
The factored form of $y^2 - y$ is:
a) $y(y)$
b) $y(y - 1)$
c) $y^2(1 - 1/y)$
d) Cannot be factored
Answer: b
Factorise $3p^2q - 12pq^2$.
a) $3pq(p - 4q)$
b) $3pq(p - 4)$
c) $12pq(3p - q)$
d) $3p(pq - 4q^2)$
Answer: a
Pull out the common factor from $18x^4 - 27x^3 + 9x^2$.
a) $9x(2x^3 - 3x^2 + x)$
b) $9x^2(2x^2 - 3x + 1)$
c) $3x^2(6x^2 - 9x + 3)$
d) $18x^2(x^2 - 1.5x + 0.5)$
Answer: b
What is the greatest common factor of $15a^3b^2$, $20a^2b^3$, and $25ab^4$?
a) $5ab$
b) $5a^2b^2$
c) $5ab^2$
d) $15ab^2$
Answer: c
Factorise completely: $-4y^3 - 16y^2 - 8y$.
a) $-4y(y^2 - 4y - 2)$
b) $-4y(y^2 + 4y + 2)$
c) $4y(-y^2 - 4y + 2)$
d) $-2y(2y^2 + 8y + 4)$
Answer: b
The factored form of $100x^5 + 50x^4$ is:
a) $50x^4(2x + 1)$
b) $100x^4(x + 0.5)$
c) $25x^4(4x + 2)$
d) $50x^5(2 + x^{-1})$
Answer: a

2. Factorising by Grouping

Method: Used when an expression has four or more terms. Group terms into pairs that share a common factor, factor out the GCF from each pair, and then factor out the common binomial.

Numerical Example:
Factorise $ax + ay + bx + by$.
Group: $(ax + ay) + (bx + by)$.
Factor pairs: $a(x + y) + b(x + y)$.
Factor out binomial: $(x + y)(a + b)$.

MCQs - Grouping

Grouping is most commonly used for polynomials with how many terms?
a) 2
b) 3
c) 4
d) 1
Answer: c
Factorise $x^3 + x^2 + x + 1$ by grouping.
a) $(x^2 + 1)(x + 1)$
b) $x^2(x + 1)$
c) $(x + 1)^3$
d) $(x^2 - 1)(x - 1)$
Answer: a (Group: $x^2(x+1) + 1(x+1)$)
Factorise $2xy - 6y + x - 3$.
a) $(2y - 1)(x - 3)$
b) $(2y + 1)(x - 3)$
c) $(y + 2)(x - 3)$
d) $(2y + 1)(x + 3)$
Answer: b (Group: $2y(x-3) + 1(x-3)$)
What is the common binomial factor when $ab + ac - b - c$ is grouped as $a(b+c) - 1(b+c)$?
a) $a-1$
b) $a+1$
c) $b+c$
d) $b-c$
Answer: c
Factorise $m^2 - mn + 4m - 4n$.
a) $(m - n)(m - 4)$
b) $(m + n)(m + 4)$
c) $(m - n)(m + 4)$
d) $(m + n)(m - 4)$
Answer: c ($m(m-n) + 4(m-n)$)
Factorise $ax + bx - ay - by$.
a) $(x - y)(a - b)$
b) $(x + y)(a - b)$
c) $(x - y)(a + b)$
d) $(x + y)(a + b)$
Answer: c
Factorise $2p^3 + p^2 + 8p + 4$.
a) $(p^2 + 4)(2p + 1)$
b) $(p^2 + 2)(2p + 1)$
c) $(p^2 + 4)(p + 2)$
d) $(2p^2 + 4)(p + 1)$
Answer: a
Factorise $5x^2 - 20x + x - 4$.
a) $(5x - 1)(x - 4)$
b) $(5x + 1)(x - 4)$
c) $(5x + 1)(x + 4)$
d) $(x - 1)(5x - 4)$
Answer: b
The expression $x^3 - 3x^2 - 2x + 6$ can be factored into:
a) $(x^2 - 2)(x + 3)$
b) $(x^2 + 2)(x - 3)$
c) $(x^2 - 2)(x - 3)$
d) $(x^2 - 3)(x - 2)$
Answer: c
Which polynomial can be factored as $(3y - 2)(y^2 + 5)$?
a) $3y^3 - 2y^2 + 15y - 10$
b) $3y^3 + 2y^2 - 15y + 10$
c) $3y^3 - 2y^2 - 15y + 10$
d) $3y^3 + 2y^2 + 15y + 10$
Answer: a

3. Factorising using Identities

Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$
Perfect Square Trinomials:
- $a^2 + 2ab + b^2 = (a + b)^2$
- $a^2 - 2ab + b^2 = (a - b)^2$

Numerical Example:
Factorise $4x^2 - 25$.
Recognize it as $(2x)^2 - (5)^2$.
Apply difference of squares: $(2x - 5)(2x + 5)$.

MCQs - Using Identities

Factorise $x^2 - 16$.
a) $(x-4)^2$
b) $(x-16)(x+1)$
c) $(x-4)(x+4)$
d) $(x+4)^2$
Answer: c
The expression $x^2 + 10x + 25$ factors into:
a) $(x+5)(x-5)$
b) $(x+5)^2$
c) $(x-5)^2$
d) $(x+10)^2$
Answer: b
Factorise $9y^2 - 49$.
a) $(3y - 7)(3y + 7)$
b) $(9y - 7)(y + 7)$
c) $(3y - 49)(3y + 1)$
d) $(3y - 7)^2$
Answer: a
Which of the following is a perfect square trinomial?
a) $x^2 + 6x + 8$
b) $x^2 - 8x + 16$
c) $x^2 + 4x + 16$
d) $x^2 - 9$
Answer: b (Because it factors to $(x-4)^2$)
Factorise $1 - 36x^2$.
a) $(1 - 6x)^2$
b) $(1 + 6x)^2$
c) $(1 - 6x)(1 + 6x)$
d) $(1 - 36x)(1 + x)$
Answer: c
Factorise $49a^2 - 64b^2$.
a) $(7a - 8b)^2$
b) $(7a - 8b)(7a + 8b)$
c) $(49a - 64b)(a + b)$
d) $(7a + 8b)^2$
Answer: b
The factored form of $16x^2 + 24x + 9$ is:
a) $(4x + 3)(4x - 3)$
b) $(8x + 3)^2$
c) $(4x + 3)^2$
d) $(16x + 9)^2$
Answer: c
Factorise $25 - 10y + y^2$.
a) $(5 - y)(5 + y)$
b) $(y - 5)^2$
c) $(y + 5)^2$
d) $(25 - y)^2$
Answer: b
Which expression is equivalent to $(x^2 - 81)$?
a) $(x - 9)^2$
b) $(x + 9)^2$
c) $(x - 9)(x + 9)$
d) $(x - 81)(x + 1)$
Answer: c
Factorise $100x^2 - 121y^2$.
a) $(10x - 11y)(10x + 11y)$
b) $(10x - 11y)^2$
c) $(100x - 121y)(x + y)$
d) $(10x + 11y)^2$
Answer: a

4. Factorising Quadratic Trinomials ($x^2 + bx + c$)

Method: Find two numbers that multiply to $c$ and add up to $b$. Let's call them $p$ and $q$.
The factored form is $(x + p)(x + q)$.

Numerical Example:
Factorise $x^2 + 7x + 12$.
Multiply to $12$: (1,12), (2,6), (3,4).
Add to $7$: $3 + 4 = 7$.
So, $p=3$ and $q=4$. Factored form: $(x + 3)(x + 4)$.

MCQs - Quadratic Trinomials

To factor $x^2 - 5x + 6$, you need numbers that multiply to 6 and add to:
a) 6
b) 5
c) -5
d) 1
Answer: c
Factorise $x^2 + 8x + 15$.
a) $(x+3)(x+5)$
b) $(x-3)(x-5)$
c) $(x+1)(x+15)$
d) $(x+8)(x+15)$
Answer: a
Factorise $x^2 - 2x - 8$.
a) $(x-4)(x+2)$
b) $(x+4)(x-2)$
c) $(x-4)(x-2)$
d) $(x+4)(x+2)$
Answer: a ($-4 \times 2 = -8$, and $-4 + 2 = -2$)
Which of the following trinomials cannot be factored using integer values?
a) $x^2 + 4x + 4$
b) $x^2 + x + 1$
c) $x^2 + 3x + 2$
d) $x^2 - x - 2$
Answer: b (No integers multiply to 1 and add to 1)
The factors of $y^2 - 10y + 24$ are:
a) $(y - 12)(y + 2)$
b) $(y + 6)(y + 4)$
c) $(y - 6)(y - 4)$
d) $(y - 8)(y - 3)$
Answer: c
Factorise $x^2 - 9x + 18$.
a) $(x - 3)(x - 6)$
b) $(x + 3)(x + 6)$
c) $(x - 2)(x - 9)$
d) $(x + 2)(x + 9)$
Answer: a
To factor $x^2 + 2x - 15$, you need numbers that multiply to -15 and add to:
a) 2
b) -2
c) -15
d) 15
Answer: a
Factorise $a^2 - a - 20$.
a) $(a - 5)(a + 4)$
b) $(a + 5)(a - 4)$
c) $(a - 10)(a + 2)$
d) $(a + 10)(a - 2)$
Answer: a
The factors of $y^2 + 11y + 24$ are:
a) $(y + 12)(y + 2)$
b) $(y + 8)(y + 3)$
c) $(y + 6)(y + 4)$
d) $(y + 24)(y + 1)$
Answer: b
Which trinomial has $(x - 7)(x + 2)$ as its factored form?
a) $x^2 + 5x - 14$
b) $x^2 - 5x + 14$
c) $x^2 - 5x - 14$
d) $x^2 + 9x - 14$
Answer: c

Fun Facts about Factorisation!

Prime Polynomials: Just like prime numbers cannot be factored into smaller integers, some polynomials cannot be factored into smaller polynomials. These are called "irreducible" or "prime" polynomials.
Cryptography: Modern digital security (like RSA encryption) relies on the fact that while multiplying two massive prime numbers is easy for a computer, factorising the result back into those primes is practically impossible!

Factorisation Of Algebraic Expressions