Using Patterns to Multiply Two Binomials
We obtain another useful pattern when we multiply the sum and difference
of the same two terms.
| For example, let’s use FOIL
to find
(a + b)(a - b). |
(a + b)(a - b) |
= |
F a2 |
- |
O ab |
+ |
I ba |
- |
L b2 |
| When we combine like terms,
the middle terms cancel out. |
|
= a2 - b2 |
The expression a2 - b2 is called the difference of two squares
because the operation is subtraction and each term is a square.
The pattern tells us that the product of conjugates always results in the
difference of two perfect squares.
Formula —
The Product of the Sum and Difference of the Same Two Terms
Let a and b represent any real numbers.
(a + b)(a - b) = a2 - b2
Note:
The following pairs of binomials are
examples of conjugates:
a + b and a - b
x - y and x + y
w + 2 and w - 2
k - 5 and k + 5
Example
Find: (m + 8n)(m - 8n)
Solution
| The expression (m + 8n)(m - 8n) is in the form (a
+ b)(a - b). So we
can use the shortcut:
|
(a + b)(a - b) |
= a2 - b2 |
| Substitute m for a and 8n for b. Simplify.
So, (m + 8n)(m - 8n) = m2 - 64n2. |
(m + 8n)(m - 8n) |
= (m)2 - (8n)2 = m2 -
64n2 |
|
