Chapter 3 Lesson Notes: Expansion and Factorisation of Algebraic Expressions

Expansion of Algebraic Expressions

Expansion is the process of removing brackets in an algebraic expression by multiplying out the terms. The goal is to rewrite the expression as a sum or difference of terms.

Distributive Law

This law states that a(b+c) = ab + ac.
Each term inside the parentheses is multiplied by the term outside.

Example:

3(x+5) = 3x + 15

-2(y-4) = -2y + 8

Multiplying Two Monomials
Definition : A monomial is a single term consisting of a coefficient (a number) multiplied by one or more variables raised to non-negative integer powers.

Multiply the coefficients.
Add the powers of the same variables.

Example:

(3x)(5x^2) = (3 \times 5)(x^{1+2}) = 15x^3

Multiplying a Monomial by a Polynomial
Definition: A polynomial is an expression that consists of one or more monomials combined using addition or subtraction.

Apply the distributive law to each term within the polynomial.

Example:

2x(x^2 - 3x + 4) = 2x(x^2) - 2x(3x) + 2x(4) = 2x^3 - 6x^2 + 8x

Multiplying Two Binomials
Definition: A binomial is an algebraic expression that consists of exactly two terms, combined by addition or subtraction.

Method (FOIL):

First: Multiply the first terms of each binomial.
Outer: Multiply the outer terms.
Inner: Multiply the inner terms.
Last: Multiply the last terms of each binomial.

Combine like terms after multiplication.

Example:

(x+3)(x+2) = x(x) + x(2) + 3(x) + 3(2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6

Special Algebraic Identities (Special Products)

These are common expansion patterns that should be memorized:

Square of a Sum:
(a+b)^2 = a^2 + 2ab + b^2
Square of a Difference:
(a-b)^2 = a^2 - 2ab + b^2
Difference of Squares:
(a+b)(a-b) = a^2 - b^2

Example:

(2x+3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9

(y-5)^2 = y^2 - 2(y)(5) + 5^2 = y^2 - 10y + 25

(4p+q)(4p-q) = (4p)^2 - q^2 = 16p^2 - q^2

Factorisation of Algebraic Expressions

the reverse process of expansion.
involves rewriting an algebraic expression as a product of its factors
The goal is often to factorise the expression to make it simpler for solving equations or simplifying mathematical problems.

Factorising by Taking Out a Common Factor

Identify the greatest common factor (GCF) of all terms in the expression.
Divide each term by the GCF and place the GCF outside the parentheses.

Example:

5x + 10 = 5(x+2)

6x^2y - 9xy^2 = 3xy(2x - 3y)

Factorising by Grouping (for 4 terms)

Group terms in pairs that share a common factor.
Factor out the common factor from each pair.
If a common binomial factor emerges, factor it out.

Example:

ax + ay + bx + by = a(x+y) + b(x+y) = (x+y)(a+b)

Recognize expressions that fit the patterns of the special products:

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

Example:

x^2 + 6x + 9 = (x+3)^2 (since x^2 + 2(x)(3) + 3^2)

4y^2 - 25 = (2y)^2 - 5^2 = (2y+5)(2y-5)

Factorising Quadratic Expressions (Trinomials of the form ax^2 + bx + c)

When a=1 (i.e., x^2 + bx + c):

Find two numbers that multiply to c and add up to b.

Example:

x^2 + 7x + 10

Numbers that multiply to 10: (1,10), (2,5)
Numbers that add to 7: (2,5)
Therefore, x^2 + 7x + 10 = (x+2)(x+5)

When a \neq 1 (e.g., 2x^2 + 7x + 6):

Method 1: By Grouping (Product-Sum Method)

Multiply a and c (ac).
Find two numbers that multiply to ac and add up to b.
Rewrite the middle term (bx) using these two numbers.
Factor by grouping.

Example:

2x^2 + 7x + 6

ac = 2 \times 6 = 12. (Numbers that multiply to 12 and add to 7 are 3 and 4.)

2x^2 + 4x + 3x + 6

2x(x+2) + 3(x+2)

(x+2)(2x+3)

Method 2: Trial and Error

Consider factors of a and factors of c.
Arrange them in binomials and check the sum of the inner and outer products.

Example:

2x^2 + 7x + 6

Factors of 2x^2: (2x \text{ and } x)

Factors of 6: (1,6), (2,3), (3,2), (6,1)

Try (2x+3)(x+2): Outer 4x, Inner 3x. Sum