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Simplifying Fractions 3
Factoring quadratics
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Using Patterns to Multiply Two Binomials
Adding and Subtracting Fractions
Solving Linear Inequalities
Adding Fractions
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Example 6
Dividing Monomials
Multiplication can Increase or Decrease a Number
Graphing Horizontal Lines
Simplification of Expressions Containing only Monomials
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Powers of i
Multiplying and Dividing Fractions
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Solving Equations by Factoring
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
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Division Property of Square and Cube Roots
Multiplying Two Numbers Close to but less than 100
Solving Absolute Value Inequalities
Equations of Circles
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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
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Axis of Symmetry and Vertices
Factoring Polynomials with Four Terms
Evaluation of Simple Formulas
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Scientific Notation
Lines and Equations
Horizontal and Vertical Lines
Solving Equations by Factoring
Solving Systems of Linear Inequalities
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
Algebra
Solving Quadratic Equations Using the Square Root Property
Dividing Rational Expressions
Quadratic Equations with Imaginary Solutions
Factoring Trinomials Using Patterns

Factoring Perfect Square Trinomials

The trinomial that results from squaring a binomial is called a perfect square trinomial. We can reverse the rules from Section 5.4 for the square of a sum or a difference to get rules for factoring.

 

Factoring Perfect Square Trinomials

a2 + 2ab + b2 = (a + b)2

a2 - 2ab + b2 = (a - b)2

Consider the polynomial x2 + 6x + 9. If we recognize that

x2 + 6x + 9 = x2 + 2 · x · 3 + 32,

then we can see that it is a perfect square trinomial. It fits the rule if a = x and b = 3:

x2 + 6x + 9 = (x - 3)2

Perfect square trinomials can be identified by using the following strategy.

 

Strategy for Identifying Perfect Square Trinomials

A trinomial is a perfect square trinomial if

1. the first and last terms are of the form a2 and b2,

2. the middle term is 2 or -2 times the product of a and b.

We use this strategy in the next example.

 

Example 1

Factoring perfect square trinomials

Factor each polynomial.

a) x2 - 8x + 16

b) a2 + 14a + 49

c) 4x2 + 12x + 9

Solution

a) Because the first term is x2, the last is 42, and -2(x)(4) is equal to the middle term -8x, the trinomial x2 - 8x + 16 is a perfect square trinomial:

x2 - 8x + 16 = (x - 4)2

b) Because 49 = 72 and 14a = 2(a)(7), we have a perfect square trinomial:

a2 + 14a + 49 = (a + 7)2

c) Because 4x2 = (2x)2, 9 = 32, and the middle term 12x is equal to 2(2x)(3), the trinomial 4x2 + 12x + 9 is a perfect square trinomial:

4x2 + 12x + 9 = (2x + 3)2

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