Using Patterns to Multiply Two Binomials
We can use FOIL to find the product of any two binomials. Sometimes we
can find certain binomial products more quickly by recognizing patterns.
For example, let’s first use FOIL to find (a + b)2.
| Use the definition of exponential notation to write (a +
b)2 as the product of two binomials. |
(a + b)2 |
= (a + b)(a + b) |
| Use FOIL.
Combine like terms.
Thus, (a + b)2 = a2 + 2ab + b2 |
|
= a2 + ab + ba + b2
= a2 + 2ab + b2 |
Note:
Note that (a + b)2 is not equal to a2 + b2. Don’t forget the middle term, 2ab.
We obtain a similar pattern when we square a binomial that is a difference
rather than a sum.
Formula —
Square of a Binomial
Let a and b represent any real numbers.
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
When a binomial is squared, the resulting trinomial is called a
perfect square trinomial.
For example, both a2 + 2ab + b2 and a2 - 2ab + b2 are perfect square
trinomials.
Note:
When we refer to integers, a perfect square
is an integer that is the square of another
integer:
9 is a perfect square because it is the
result of squaring 3.
A similar situation exists for variables:
a2 is a perfect square because it is the
result of squaring a.
64n2 is a perfect square because it is the
result of squaring 8n.
Example 1
Find: (6y2 + 5)2
Solution
| The expression (6y2 + 5)2 is
in the form (a + b)2.
So, we can use the formula
for the square of a binomial.
|
(a + b)2 |
= a2 + 2ab + b2 |
| Substitute 6y2 for a and 5 for b.
Simplify.
So, (6y2 + 5)2 = 36y4 + 60y2
+ 25 |
(6y2 + 5)2
|
= (6y2)2 + 2(6y2)5
+ (5)2 = 36y4 + 60y2 + 25 |
Note that (6y2 + 5)2 ≠ (6y2)2
+ (5)2. Don’t forget the middle term, 60y2.
Example 2
Find: (3w - 7y)(3w - 7y)
Solution
| Since (3w - 7y)(3w - 7y) = (3w - 7y)2, we can use the shortcut for the square of a binomial. |
(a - b)2 |
= a2 - 2ab + b2 |
| Substitute 3w for aand
7y for b.
Simplify. |
(3w - 7y)2
|
= (3w)2 + 2(3w)(7y)
+ (7y)2 = 90w2 - 42wy + 49y2 |
| So, (3w - 7y)(3w - 7y) = 90w2 - 42wy + 49y2. |
Note that (3w2 - 7y)2 ≠ (3w2)2
- (7y)2. Don’t forget the middle term, -42wy2.
|
