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REVISION: ALGEBRAIC EXPRESSIONS AND EXPONENTS
11 MARCH
Lesson Description
In this lesson we revise how to:





Expand algebraic expressions
Factorise algebraic expressions
Simplify algebraic fractions
Simplify expressions with rational exponents
Solve exponential equations
Key Concepts
Products
A product is formed when two or more numbers or algebraic terms are multiplied together.
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
2
= 2x - 4x + 2x -3x + 6x -3
3
2
= 2x - 7x + 8x -3
Question 2 (Products)
Expand the following trinomials:
a.)
(2x − 2)(3x + 8)
b.)
(3 – 4x)(2 + 3x)
Question 3 (Products)
Expand the following binomials and trinomials:
2
a.)
(−2y − 4y + 11)(5y − 12)
b.)
(7y − 6y − 8)(−2y + 2)
2
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 +
2
3)
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)
Question 1 (Factorise)
Factorise the following:
a.)
b.)
2
x – 2x − ax + 2x
2
6x − 15x − 9
Simplifying Algebraic Fractions
The same rules for multiplying, dividing, adding and subtracting fractions that you have practiced
since primary school apply to algebraic fractions. The first step in any algebraic fraction problem is to
identify all the factors of the denominators and numerators. You must factorise to simplify you
problems.
Rule for multiplication of algebraic fractions
You can multiply the numerators by the numerators and the denominators by denominators.
Before expanding try to simplify by cancelling out any common factors in the numerator and
denominator. Apply the distributive law to expand uncommon factors.
Example:
Rule for division of algebraic fractions
Remember that dividing by a fraction is the same as multiplying by the reciprocal. You change
the division sign to a multiplication and switch the denominator and numerator of the term
after the divide sign. When simplifying, make sure you factorise before expanding.
Example:
4 x2  8x
12 x
3

x2
9x
Rule for adding or subtracting algebraic fractions
To add or subtract any fraction, you must first find the lowest common denominator. You will
find this easiest if you factorise all terms first. Next you divide the LCM by each of the original
denominators. You multiply the numerators by the answer and then add or subtract the
numerators. You may need to factorise your final numerator. You can never cancel terms that
are added or subtracted. You can only cancel common factors that are on the denominator
and numerator if these are multiplied to other terms.
Example 1: When denominators are numbers
x
x

 3
2
5
x
x
3
 

2
5
1
x 5
x 2
3 10
     
2 5
5 2
1 10
5x
2 x 30



10
10 10
5 x  2 x  30

10
3 x  30

10
Example 2: When denominators have variable terms
3
5
7


4
2
2
6x
2 x y 4 xy

18 y  15 x  14 xy 2  48 x 2 y 2
12 x 2 y 2
Question 2 (Algebraic fractions)
Simplify the following algebraic expressions:
a.)
x 1
x2
x 3


2
3
4
Exponents
Definition:
= a x a x a …. to n factors of a.
Laws for exponents are restricted to bases of natural numbers.
In general,
1.
2.
3.
4.
5.
6.
Definition of a Rational Exponent
=
where a > 0 and
is a rational number
Question 1 (Exponents)
Simplify:
a)
b)
Exponential Equations
In solving exponential equations where the exponent is unknown, we will make use of the following
property:
If
then
, since base a is the same on either side of the equal sign.
Question 4 (Exponential Equations)
Solve for x
a)
b)