Consider the following sum and difference of two binomials.
(a + b) and (a - b)
Product of sum and difference of the above two binomials :
(a + b)(a - b) = (a + b)(a - b)
= a ⋅ a + a(-b) + b ⋅ a + b(-b)
= a2 - ab + ab + b2
= a2 - b2
Therefore,
(a + b)(a - b) = a2 - b2
or
a2 - b2 = (a + b)(a - b)
Example 1 :
Multiply.
(x + 5)(x - 5)
Solution :
Use the rule for (a + b)(a - b).
(a + b)(a - b) = a2 - b2
Identify a and b : a = x and b = 5.
(x + 5)(x - 5) = x2 - 52
= x2 - 25
Example 2 :
Multiply.
(x2 + 2y)(x2 - 2y)
Solution :
Use the rule for (a + b)(a - b).
(a + b)(a - b) = a2 - b2
Identify a and b : a = x2 and b = 2y.
(x2 + 2y)(x2 - 2y) = (x2)2 - (2y)2
= x4 - 4y2
Example 3 :
Multiply.
(8 + z)(8 - z)
Solution :
Use the rule for (a + b)(a - b).
(a + b)(a - b) = a2 - b2
Identify a and b : a = 8 and b = z.
(8 + z)(8 - z) = (8)2 - (z)2
= 64 - z2
Example 4 :
Multiply.
(3 + 2z2)(3 + 2z2)
Solution :
Use the rule for (a + b)(a - b).
(a + b)(a - b) = a2 - b2
Identify a and b : a = 3 and b = 2z2.
(3 + 2z2)(3 + 2z2) = (3)2 - (2z)2
= 9 - 4z2
Example 5 :
Multiply.
(a2 + b2)(a2 - b2)
Solution :
Use the rule for (a + b)(a - b).
(a + b)(a - b) = a2 - b2
Identify a and b : a = a2 and b = b2.
(a2 + b2)(a2 - b2) = (a2)2 - (b2)2
= a4 - b4
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