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Saturday 25th of February
Rotating a Parabola
Multiplying Fractions
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Simplifying Complex Fractions
Decimals and Fractions
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Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Order of Operations
Dividing Complex Numbers
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
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Factoring Out the Greatest Common Factor (GCF)

A polynomial may be factored by using division: If we know one factor of a polynomial, then we can use it as a divisor to obtain the other factor, the quotient. However, this technique is not very practical because the division process can be somewhat tedious, and it is not easy to obtain a factor to use as the divisor. In this section and the next two sections we will develop better techniques for factoring polynomials. These techniques will be used for solving equations and problems in the last section of this chapter.

A natural number larger than 1 that has no factors other than itself and 1 is called a prime number. The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23 are the first nine prime numbers.

There are infinitely many prime numbers. To factor a natural number completely means to write it as a product of prime numbers. In factoring 12 we might write 12 = 4 · 3. However, 12 is not factored completely as 4 · 3 because 4 is not a prime. To factor 12 completely, we write 12 = 2 · 2 · 3 (or 22 · 3).

We use the distributive property to multiply a monomial and a binomial:

6x(2x - 1) = 12x2 - 6x

If we start with 12x2 - 6x, we can use the distributive property to get

12x2 - 6x = 6x(2x - 1).

We have factored out 6x, which is a common factor of 12x2 and -6x. We could have factored out just 3 to get

12x2 - 6x = 3(4x2 - 2x),

but this would not be factoring out the greatest common factor. The greatest common factor (GCF) is a monomial that includes every number or variable that is a factor of all of the terms of the polynomial.


We can use the following strategy for finding the greatest common factor of a group of terms.


Strategy for Finding the Greatest Common Factor (GCF)

1. Factor each term completely.

2. Write a product using each factor that is common to all of the terms.

3. On each of these factors, use an exponent equal to the smallest exponent that appears on that factor in any of the terms.


Example 1

The greatest common factor

Find the greatest common factor (GCF) for each group of terms.

a) 8x2y, 20xy3

b) 30a2, 45a3 b2, 75a4b


a) First factor each term completely:



= 23x2y

= 22 · 5xy3

The factors common to both terms are 2, x, and y. In the GCF we use the smallest exponent that appears on each factor in either of the terms. So the GCF is 22xy or 4xy.

b) First factor each term completely:




= 2 · 3 · 5a2

32 · 5a3b2

3 · 52a4b

The GCF is 3 · 5a2 or 15a2.

To factor out the GCF from a polynomial, find the GCF for the terms, then use the distributive property to factor it out.


Example 2

Factoring out the greatest common factor

Factor each polynomial by factoring out the GCF.

a) 5x4 - 10x3 + 15x2

b) 8xy2 + 20x2y

c) 60x5 + 24x3 + 36x2


a) First factor each term completely:

5x4 = 5x4, 10x3 = 2 · 5x3, 15x2 = 3 · 5x2.

The GCF of the three terms is 5x2. Now factor 5x2 out of each term:

5x4 - 10x3 + 15x2 = 5x2(x2 - 2x + 3)

b) The GCF for 8xy2 and 20x2y is 4xy:

8xy2 + 20x2y = 4xy(2y + 5x)

c) First factor each coefficient in 60x5 + 24x3 + 36x2:

60 = 22 · 3 · 5, 24 = 23 · 3, 36 = 22 · 32.

The GCF of the three terms is 22 · 3x2 or 12x2:

60x5 + 24x3 + 36x2 = 12x2(5x3 + 2x + 3)

In the next example the common factor in each term is a binomial.


Example 3

Factoring out a binomial


a) (x + 3)w + (x + 3)a

b) x(x - 9) - 4(x - 9)


a) We treat x + 3 like a common monomial when factoring:

(x + 3)w + (x + 3)a = (x + 3)(w + a)

b) Factor out the common binomial x - 9:

x(x - 9) - 4(x - 9) = (x - 4)(x - 9)

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