Multiplying Special Binomials
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 have a sum of 0. So the product of a sum and a
difference of the same two terms is equal to the difference of two squares.
The Product of a Sum and a Difference
(a + b)(a - b) = a2 - b2
Example 1
Product of a sum and a difference
Find each product
a) (x + 2)(x - 2)
b) (b + 7)(b - 7)
c) (3x - 5)(3x + 5)
Solution
a) (x + 2)(x - 2) = x2 - 4
b) (b + 7)(b - 7) = b2 - 49
c) (3x - 5)(3x + 5) = 9x2 - 25
Higher Powers of Binomials
To find a power of a binomial that is higher thatn 2, we can use the rule for
squaring a binomial along with the method of multiplying binomials using the
distributive property. Finding the second or higher power of a binomial is
called expandin the binomial because the result has more terms than the
original.
Example 2
Higher powers of a binomial
Expand each binomial.
a) (x + 4)3
b) (y - 2)4
Solution
| a) (x + 4)3 |
= (x + 4)2(x + 4) |
| |
= (x2 + 8x + 16)(x + 4) |
| |
= (x2 + 8x + 16)x + (x2 + 8x + 16)4 |
| |
= x3 + 8x2 + 16x + 4x2 +
32x + 64 |
| |
= x3 + 12x2 + 48x + 64 |
| b) (y - 2)4 |
= (y - 2)2(y - 2)2 |
| |
= (y2 - 4y + 4)(y2 - 4y + 4) |
| |
= (y2 - 4y + 4)(y2) + (y2
- 4y + 4)(-4y) + (y2 - 4y + 4)(4) |
| |
= y4 - 4y3 + 4y2 - 4y3
+ 16y2 - 16y + 4y2 - 16y + 16 |
| |
= y4 - 8y3 + 24y2 + 16 |
|
