From the multiplication rules for sums and polynomials can easily obtain the following seven special factors.
They should know by heart, as they are used in all problems in mathematics.
[1] ( a + b )² = a² + 2ab + b² ,
[2] ( a – b )² = a² – 2ab + b² ,
[3] ( a + b ) ( a – b ) = a² – b²,
[4] ( a + b )³ = a³ + 3a² b + 3ab² + b³ ,
[5] ( a – b )³ = a ³ – 3a² b + 3ab² – b³ ,
[6] ( a + b )( a² – ab + b² ) = a³ + b³ ,
[7] ( a – b )( a ² + ab + b² ) = a³ – b³
Let’s look some examples from SAT tests :
Example 1
If (m + n)2 = 18 and mn = 4, then what is the value of m2 + n2?
Following the formula [1], and using first condition, we will get:
m2+2mn+n2 = 18,
So, if we will use second condition, we will get :
m2+8+n2 = 18, so m2 + n2 = 10.
Example 2
If (a-b) = -4, what is the value of a2-2ab + b2 ?
Following the formula [2], we will get:
a2 – 2ab + b2 = ( a – b )² = (-4)2 = 16
Example 3
If ab + 1/(ab) = 4, what is the value (ab)2 +1/(ab)2
From ab + 1/(ab) = 4 and formula [1], we will get:
(ab + 1/(ab) )2 = (ab)2 +2⨯ ab⨯ 1/(ab) + 1/(ab)2
Reducing the fraction, we will get:
(ab)2 +2+1/(ab)2 = 16, so (ab)2 +1/(ab)2 = 14
