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ALGEBRAIC EXPRESSIONS: PRODUCTS AND FACTORS (28th January
2013)
Lesson Description
In this lesson we will:

Show how to expand an algebraic expression by multiplication including:
o a binomial or trinomial by a single term
o a binomial by a binomial
o a binomial by a trinomial

Show how to factorise algebraic expression:
o by finding a common factor
o by identifying a difference of two squares
o by identifying the binomial factors of a trinomial
o by identifying a sum or difference of two cubes
o by using grouping in pairs
Key Concepts
Products
A product is formed when two or more numbers or algebraic terms are multiplied together.
When we multiply two number together, we are repeating the process of addition a certain number of
times. E.g. 3 x 4 means we must add 3 to itself 4 times (3 x 4 = 3+3+3+3)
In the same way 3a = a + a + a ( Addition of like terms)
There are three laws that apply for multiplication of numbers or algebraic terms:



Commutative law
The order of multiplication does not matter a x b = b x a
Associative Law
When more than two terms are multiplied together, the order in which you multiply does not
matter (2 x a) x b = 2 x (a x b)
Distributive Law
This law explains what happens when you add or subtract and multiply numbers or algebraic
terms together in the same expression
Example1:
4(3+5) = 4 x 8 = 32
But 4(3+5) = 4x3 +4x5 = 12 +20 = 32
Example 2:
4(5 – 2) = 4 x 3 = 12
4(5 – 2) = 4 x 5 - 4 x 2 = 20 - 8 = 12
Applying the Distributive Law to algebraic expressions
Multiplying a binomial by a number or term
Example 1: 2(x – 3) = 2x -6
Example 2: a(b + 3) = ab +3a
Multiplying a trinomial by a number or term
Example 1: 3(x + 2y + 3) = 3x +6y +9
Example 2: (a - b + 3)6 = 6a -6b + 18
Multiplying a binomial by a binomial
Example 1: (2x + y)(x -2y) = (2x.x)+ (2x.(-2y) + (y.x) + (y.(-2y)
2
2
= 2x +(-4xy) +( y. x) + (-2y )
2
2
= 2x - 4xy + x.y - 2y
2
2
= 2x - 3xy - 2y
Multiplying a binomial by a trinomial
2
2
2
Example 1: (2x-3)(x – 2x +1) = (2x. x ) + (2x. (-2x)) + (2x.(1)) –(3. x ) - (3. (-2x)) - (3.(1))
3
2
3
2
2
= 2x - 4x + 2x -3x + 6x -3
= 2x - 7x + 8x -3
Finding Factors
This is the reverse operation of expanding or finding the products. There are different patterns that
you can look for to find factors
Removing a common factor
Example
2
5x + 10yx =5x(x +2y)
Difference of two squares
Example
2
2
25x -16y = (5x - 4y)(5x + 4y)
Trinomials
Example
2
a + 2a +1 =(a + 1)(a + 1) = (a + 1)
2
b - b + 12 = (b - 4)(b +
Sum of two cubes
Example
3
x + 27 =
2
(x + 3)(x − 3x + 9)
Difference of two cubes
Example
3
2
x – 27 = (x − 3)(x + 3x + 9)
3)
2
Questions
Question 1
Simplify the following algebraic expressions by expanding fully:
a.)
-8(6y + 3)
b.)
(2 – 5p)(-6)
Question 2
Expand the following trinomials:
a.)
(2x − 2)(3x + 8)
b.)
(3 – 4x)(2 + 3x)
Question 3
Expand the following binomials and trinomials:
2
a.)
(−2y − 4y + 11)(5y − 12)
b.)
(7y − 6y − 8)(−2y + 2)
2
Question 4
Factorise by grouping in pairs:
a.)
b.)
2
x – 2x − ax + 2x
5ab – 3b + 10a − 6
Question 5
Factorise the following trinomials:
2
a.)
x + 12x + 36
b.)
6x − 15x − 9
2
Question 6
Factorise the following:
3
a.)
16y – 432
b.)
64x3 + 1
Links
2012 Learn Xtra
Algebraic Expessions
http://www.youtube.com/watch?v=J2HHXvOurmU
Everything Maths
Chapter 1
ALGEBRAIC EXPRESSIONS
http://everythingmaths.co.za/grade-10/maths/grade-10/everything-maths-grade-10.pdf
Algebra Worksheets
http://www.math-drills.com/algebra.shtml