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Fields Medal Prize Winners (1998)




TUTORIALS:


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Fractions
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Roots
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Powers
Rewriting Algebraic Fractions
Exponents
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Simplifying Fractions

An indispensable aspect of any work with algebraic fractions is simplification of the result. We will demonstrate techniques for simplification of fractions throughout these notes where the need arises, but in this specific document, we’ll set the stage by working through a number of examples, focussing primarily on the strategies for simplification themselves. Despite the differences in appearance of the examples to follow, there really is just one approach available for simplifying fractions:

(i) factor the numerator and denominator as completely as possible, and then

(ii) cancel any factors the numerator and denominator have in common.

If you take care to always do step (i) as a separate explicit step of work, you will rarely go wrong in simplifying fractions. What follows in this document is a series of quite a few worked out examples. They tend to be quite repetitive because there is just the one simple strategy just described, which must be applied to each one. Don’t just read through the rest of this document quickly to notice how easily we’ve worked out the solutions (and perhaps to notice how boring this sort of stuff can become). Instead, after studying the first few examples carefully, use the remaining examples as practice problems: cover up the solutions and try to work them out on your own. When you are done in each case, compare your method and final result with the one we’ve given.

 

Example 1:

Simplify .

solution:

The numerator and denominator of this fraction are already nearly fully factored. Using brackets to make these factors explicit, we get

as the final result.

 

Example 2:

Simplify .

solution:

Writing the numerator and denominator with the factors explicitly separated gives

 

Example 3:

Simplify .

solution:

The denominator is factored, but the numerator is not factored. Using the stepwise approach for factoring illustrated earlier in these notes, we get

Thus

as the final result. There is no need to multiply this last form out to remove the brackets, unless there had been some specific instruction to do so.

Note that the first step of factoring is absolutely essential here in order to get a correct result – a simpler mathematical expression which is mathematically equivalent to the original expression. A common error is to just focus on one of the terms in the numerator:

This “simplified” expression is not equivalent to the original fraction, and so an error has been made. You can easily demonstrate this by doing a test calculation:

but

ince for this one value of x, the expression 20x – 45x 3 does not give the same value that the original fraction gives, we know that 20x – 45x 3 cannot be a correct simplification of the original fraction. The error came from the fact that we cancelled the 3x of the denominator into just one of the terms in the numerator. Instead, a valid cancellation must be against factors of the entire numerator, as was done in the original solution of this example.