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  Wednesday 19th of September      
 
   
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Scientific Notation
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Linear Equations and Inequalities in One Variable
Graphing Equations in Three Variables
What the Standard Form of a Quadratic can tell you about the graph
Simplifying Radical Expressions Containing One Term
Adding and Subtracting Fractions
Multiplying Radical Expressions
Adding and Subtracting Fractions
Multiplying and Dividing With Square Roots
Graphing Linear Inequalities
Absolute Value Function
Real Numbers and the Real Line
Monomial Factors
Raising an Exponential Expression to a Power
Rational Exponents
Multiplying Two Fractions Whose Numerators Are Both 1
Multiplying Rational Expressions
Building Up the Denominator
Adding and Subtracting Decimals
Solving Quadratic Equations
Scientific Notation
Like Radical Terms
Graphing Parabolas
Subtracting Reverses
Solving Linear Equations
Dividing Rational Expressions
Complex Numbers
Solving Linear Inequalities
Working with Fractions
Graphing Linear Equations
Simplifying Expressions That Contain Negative Exponents
Rationalizing the Denominator
Decimals
Estimating Sums and Differences of Mixed Numbers
Algebraic Fractions
Simplifying Rational Expressions
Linear Equations
Dividing Complex Numbers
Simplifying Square Roots That Contain Variables
Simplifying Radicals Involving Variables
Compound Inequalities
Factoring Special Quadratic Polynomials
Simplifying Complex Fractions
Rules for Exponents
Finding Logarithms
Multiplying Polynomials
Using Coordinates to Find Slope
Variables and Expressions
Dividing Radicals
Using Proportions and Cross
Solving Equations with Radicals and Exponents
Natural Logs
The Addition Method
Equations
   
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Algebraic Fractions

Algebraic fractions have properties which are the same as those for numerical fractions, the only difference being that the the numerator (top) and denominator (bottom) are both algebraic expressions.

Example 1

Simplify each of the following fractions.

Solution

N.B. The cancellation in (b) is allowed since x is a common factor of the numerator and the denominator .

Sometimes a little more work is necessary before an algebraic fraction can be reduced to a simpler form.

Example 2

Simplify the algebraic fraction

Solution

In this case the numerator and denominator can be factored into two terms, thus

With this established the simplification proceeds as follows:

(cancelling (x-1))

Exercise 1

Simplify each of the following algebraic fractions.

(a)

(b)

Solution

(a) The fraction is . This time, instead of expanding the factors, it is easier to use the rule for powers

.

Thus

(b) In this case, some initial factorisation is needed.

Thus

where the factor ( y + 5) has been cancelled.

 

Quiz

Which of the following is a simplified version of

Solution

The numerator and denominator respectively factorise as

so that

where the factor ( t - 1) has been cancelled from the first equation.

 

So far, simplification has been achieved by cancelling common factors from the numerator and denominator. There are fractions which can be simplified by multiplying the numerator and denominator by an appropriate common factor, thus obtaining an equivalent, simpler expression.

Example 3

Simplify the following fractions.

Solution

(a) In this case, multiplying both the numerator and the denominator by 4 gives:

(b) To simplify this expression, multiply the numerator and denominator by {\f2 x. Thus

Exercise 2

Simplify each of the following algebraic fractions.

(a)

(b)

Solution

(a) The fraction is simplified by multiplying both the numerator and the denominator by 2 .

(b) In this case, since the numerator contains the fraction 1/3 and the denominator contains the fraction 1/2 , the common factor needed is 2 × 3 = 6 . Thus

Quiz

Which of the following is a simplified version of

Solution

For , the common multiplier is ( x + 1) . Multiplying the numerator and the denominator by this gives:

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