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Multiplying Binomials Using the FOIL Method
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Algebra
Order of Operations
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The Appearance of a Polynomial Equation
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Positive Integral Divisors
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Use of Parentheses or Brackets (The Distributive Law)
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Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
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Writing the Terms of a Polynomial in Descending Order
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Intercepts of a Line
Completing the Square
Order of Operations
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Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
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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
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Multiplying and Dividing Complex Numbers
Properties of Exponents
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Radicals
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
   

Product of a Sum and a Difference

If we multiply the sum a + b and the difference a - b by using FOIL, we get

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

The inner and outer products add up to zero, canceling each other out. So the product of a sum and a difference is the difference of two squares, as shown in the following rule.

 

Rule for the Product of a Sum and a Difference

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

 

Example 1

Finding the product of a sum and a difference

Find the products.

a) (x + 3)(x - 3)

b) (a3 + 8)(a3 - 8)

c) (3x2 - y3)(3x2 + y3)

Solution

a) (x + 3)(x - 3) = x2 - 9

b) (a3 + 8)(a3 - 8) = a6 - 64

c) (3x2 - y3)(3x2 + y3) = 9x4 - y6

Helpful hint

You can use (a + b)(a - b) = a2 - b2 to perform mental arithmetic tricks such as 59 · 61 = 3600 - 1 = 3599. What is 49 · 51? 28 · 32?

The square of a sum, the square of a difference, and the product of a sum and a difference are referred to as special products. Although the special products can be found by using the distributive property or FOIL, they occur so frequently in algebra that it is essential to learn the new rules. In the next example we use the special product rules to multiply two trinomials and to square a trinomial.

 

Example 2

Using special product rules to multiply trinomials

Find the products.

a) [(x + y) + 3][(x + y) - 3]

b) [(m - n) + 5]2

Solution

a) Use the rule (a + b)(a - b) = a2 - b2 with a = x + y and b = 3:

[(x + y) + 3][(x + y) - 3] = (x + y)2 - 32
  = x2 +2xy + y2 - 9

b) Use the rule (a + b)2 = a2 + 2ab + b2 with a = m - n and b = 5:

[(m - n) + 5]2 = (m - n)2 + 2(m - n)5 + 52
  = m2 - 2mn + n2 + 10m - 10n + 25

 

 
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