Fields Medal Prize Winners (1998)


Adding and Subtracting Monomials
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
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
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
Rewriting Algebraic Fractions
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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Finding the Least Common Denominator

To add fractions with denominators that are not identical, we use the basic principle of rational numbers to build up the denominators to the least common denominator (LCD). For example,

The LCD 12 is the least common multiple (LCM) of the numbers 4 and 6.

Finding the LCM for a pair of large numbers such as 24 and 126 will help you to understand the procedure for finding the LCM for any polynomials. First factor the numbers completely:

24 = 23 · 3
126 = 2 · 32 · 7

Any number that is a multiple of both 24 and 126 must have all of the factors of 24 and all of the factors of 126 in its factored form. So in the LCM we use the factors 2, 3, and 7, and for each factor we use the highest power that appears on that factor. The highest power of 2 is 3, the highest power of 3 is 2, and the highest power of 7 is 1. So the LCM is 23 · 32 · 7. If we write this product without exponents, we can see clearly that it is a multiple of both 24 and 126:

The strategy for finding the LCM for a group of polynomials can be stated as follows.


Strategy for Finding the LCM for Polynomials

1. Factor each polynomial completely. Use exponents to express repeated factors.

2. Write the product of all of the different factors that appear in the polynomials.

3. For each factor, use the highest power of that factor in any of the polynomials.


Helpful hint

The product of 24 and 126 is 3024 and 3024 is a common multiple but not the least common multiple of 24 and 126. If you divide 3024 by 6, the greatest common factor of 24 and 126, you get 504.


Example 1

Finding the LCM

Find the least common multiple for each group of polynomials.

a) 4x2y, 6y

b) a2bc, ab3c2, a3bc

c) x2 + 5x + 6, x2 + 6x + 9


a) Factor 4x2y and 6y as follows:

4x2y = 22 · x2y, 6y = 2 · 3y

To get the LCM, we use 2, 3, x, and y the maximum number of times that each appears in either of the expressions. The LCM is 22 · 3 · x2y, or 12x2y.

b) The expressions a2bc, ab3c2, and a3bc are already factored. To get the LCM, we use a, b, and c the maximum number of times that each appears in any of the expressions. The LCM is a3b3c2.

c) Factor x2 + 5x + 6 and x2 + 6x + 9 completely:

x2 + 5x + 6 = (x + 2)(x + 3), x2 + 6x + 9 = (x + 3)2

The LCM is (x + 2)(x + 3)2.