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The
following are special multiplications involving binomials that you will want to
try to remember.
Be sure to notice the patterns in each situation. You will be seeing these
patterns in numerous problems.
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Don't panic! If you cannot remember
these patterns, you can arrive at your answer by simply
multiplying with the distributive method. These
patterns are, however, very popular. If you can remember
the patterns, you can save yourself some work.
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Let's examine
these patterns:
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Squaring
a Binomial -
multiplying times
itself
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
Notice the middle
terms in both of these problems. In each problem, the middle term is
twice
the multiplication of the terms used to create the
binomial expression.
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Example 1: (x + 3)² |
= (x + 3)(x + 3) |
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= x² + 3x + 3x + 9
Distributive method |
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= x² + 6x + 9 |
* Notice the middle term.
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Example 2:
(x - 4)² |
= (x - 4)(x
- 4) |
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= x² - 4x - 4x + 16
Distributive method |
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= x² - 8x + 16 |
* Again, notice the middle
term.
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Product
of Sum and Difference
(notice
that the binomials differ only by the sign between the terms)
(a +
b)(a - b) = a² - b²
Notice that there appears to be no "middle" term
to forma a trinomial answer, as was seen in the problems above. When
multiplication occurs, the values that would form the
middle term of a trinomial actually add to zero.
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Example 3:
(x + 3)(x - 3) |
= x² - 3x + 3x - 9
Distributive method
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= x² - 9
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*Notice how the middle term
is zero.
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(2x + 3y)(2x - 3y) |
= 4x² - 6xy + 6xy - 9y²
Distributive
method |
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= 4x² - 9y²
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* Again, notice how the
middle term is zero.
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