Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Factoring Polynomials

We now begin a brief discussion of the algebraic operation called factoring. To factor an expression means to rewrite it entirely as a product. Thus, for example, by the time we’re done, you will be able to determine that

3x 4 y + 6x 3 y - 45x 2 y

can be rewritten equivalently as

3x 2 y (x + 5)(x - 3)

Note that this second expression is a product of a monomial factor, 3x 2 y, and two binomial factors. It has just one term, which is the product of these factors. If it had more than one term, it would not be in factored form.

Factoring is one of the topics in algebra that many people remember with nearly the least fondness. It seems to be a lot of tedious work with no apparent purpose. However, as we will demonstrate throughout these notes, the ability to factor at least simple algebraic expressions that can be factored, can make it possible to simplify algebraic fractions (which are even worse to deal with than factoring, surely) and to rearrange or manipulate formulas (which is a very important skill in many technical applications), among other things. No matter what people may tell you, a basic skill in factoring at least simple algebraic expressions is an important tool in technologies that requires some use of mathematics.

Not all algebraic expressions can be factored. Instead, perhaps most of them cannot be factored. The method we’ll describe in several steps in the notes to follow will let you determine systematically and quickly whether or not an expression can be factored, and if it can be factored, the method will produce that factorization as part of the process.

(By the way, you should be able to verify that the first expression above can be obtained by expanding the second expression, using methods already described in these notes, and so you should be able to verify that the two expressions are mathematically equivalent. You might proceed as follows. First, multiply the two binomials together and simplify:

(x + 5)(x - 3) = (x + 5)x + (x + 5)(-3)

= (x)(x + 5) + (-3)(x + 5)

= (x)(x) + (x)(5) + (-3)(x) + (-3)(5)

= x 2 + 5x - 3x - 15

= x 2 + 2x - 15


3x 2 y (x + 5)(x - 3) = 3x 2 y (x 2 + 2x - 15)

= 3x 2 y (x 2 ) + 3x 2 y (2x) + 3x 2 y (-15)

= 3x 4 y + 6x 3 y - 45x 2 y

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