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Arithmetic Operations with Numerical Fractions
Multiplying a Polynomial by a Monomial
Solving Linear Equation
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Simplifying Fractions 3
Factoring quadratics
Special Products
Writing Fractions as Percents
Using Patterns to Multiply Two Binomials
Adding and Subtracting Fractions
Solving Linear Inequalities
Adding Fractions
Solving Systems of Equations -
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Integer Exponents
Example 6
Dividing Monomials
Multiplication can Increase or Decrease a Number
Graphing Horizontal Lines
Simplification of Expressions Containing only Monomials
Decimal Numbers
Negative Numbers
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Adding and Subtracting Fractions
Powers of i
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Finding the Coordinates of a Point
Fractions and Decimals
Rational Expressions
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Slope of a Line
Percent Introduced
Reducing Rational Expressions to Lowest Terms
The Hyperbola
Standard Form for the Equation of a Line
Multiplication by 75
Solving Quadratic Equations Using the Quadratic Formula
Raising a Product to a Power
Solving Equations with Log Terms on Each Side
Monomial Factors
Solving Inequalities with Fractions and Parentheses
Division Property of Square and Cube Roots
Multiplying Two Numbers Close to but less than 100
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Equations of Circles
Percents and Decimals
Integral Exponents
Linear Equations - Positive and Negative Slopes
Multiplying Radicals
Factoring Special Quadratic Polynomials
Simplifying Rational Expressions
Adding and Subtracting Unlike Fractions
Graphuing Linear Inequalities
Linear Functions
Solving Quadratic Equations by Using the Quadratic Formula
Adding and Subtracting Polynomials
Adding and Subtracting Functions
Basic Algebraic Operations and Simplification
Simplifying Complex Fractions
Axis of Symmetry and Vertices
Factoring Polynomials with Four Terms
Evaluation of Simple Formulas
Graphing Systems of Equations
Scientific Notation
Lines and Equations
Horizontal and Vertical Lines
Solving Equations by Factoring
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Adding and Subtracting Rational Expressions with Different Denominators
Adding and Subtracting Fractions
Solving Linear Equations
Simple Trinomials as Products of Binomials
Solving Nonlinear Equations by Factoring
Solving System of Equations
Exponential Functions
Computing the Area of Circles
The Standard Form of a Quadratic Equation
The Discriminant
Dividing Monomials Using the Quotient Rule
Squaring a Difference
Changing the Sign of an Exponent
Adding Fractions
Powers of Radical Expressions
Steps for Solving Linear Equations
Quadratic Expressions Complete Squares
Fractions 1
Properties of Negative Exponents
Factoring Perfect Square Trinomials
Solving Quadratic Equations Using the Square Root Property
Dividing Rational Expressions
Quadratic Equations with Imaginary Solutions
Factoring Trinomials Using Patterns
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Simplifying Fractions

Examples with solutions


Example 1:

Simplify .


There is no possibility of factoring the denominator further here, since neither term in the denominator is a product. The numerator is a trinomial, and so possibly can be factored into a product of two binomials:

As explained earlier in these notes, for this to be possible, we would need to find numbers ‘a’ and ‘b’ such that a + b = -5 and ab = 6. You have probably already noticed that a = -2 and b = -3 will work. It is easy to confirm that

(x – 3)(x – 2) = x 2 – 5x + 6.

So, our original fraction becomes

in simplest form.


Example 2:

Simplify .


Proceeding as in the previous example, we quickly find that

At first, it may not appear that any cancellation is possible because neither of the factors in the numerator look the same as the denominator. However, recall that (x – 5) = -(5 – x). So, we can write our fraction here as

since now the presence of the common factor (5 – x) in the numerator and denominator is obvious. Cancelling that common factor leads to the final result.


Example 3:

Simplify .


By now, you are very familiar with the process: factor the numerator and denominator and cancel common factors. Here this gives

as the final answer.


Example 4:

Simplify .


Factoring and taking into account for the minus sign in the denominator gives

as the final simplified result.

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