Examples of a3-b3
You might already be familiar with the proof of this formula. Here's a detailed explanation:
We start with the expansion of ((a - b)3):
((a - b)3 = a3 - 3a2b + 3ab2 - b3
We can rearrange this to express (a3 - b3):
(a3 - b3 = (a - b)3 + 3a2b - 3ab2
Factoring out ((a - b)) from the right-hand side, we get:
(a3 - b3 = (a - b)((a - b)2 + 3ab)
Next, simplify the expression inside the parentheses:
((a - b)2 = a2 - 2ab + b2
Therefore:
(a3 - b3 = (a - b)(a2 - 2ab + b2 + 3ab)
Combining like terms, we get:
(a3 - b3 = (a - b)(a2 + ab + b2)
Thus, we have proven that:
(a3 - b3 = (a - b)(a2 + ab + b2)
Example 1: Verification with (a = 4) and (b = 2)
Let's verify the formula with (a = 4) and (b = 2):
((43 - 23) = (4 - 2)(42 + 4 cdot 2 + 22)
Calculate each term:
LHS = 64 - 8 = 56
RHS = 2 cdot (16 + 8 + 4) = 2 cdot 28 = 56
Both sides are equal, confirming the identity.
Example 2: Calculating (33 - 23)
Let's find the value of (33 - 23):
((33 - 23) = (3 - 2)(32 + 3 cdot 2 + 22)
Calculate each term:
LHS = 27 - 8 = 19
RHS = 1 cdot (9 + 6 + 4) = 1 cdot 19 = 19
Again, both sides are equal.
Other Important Identities
Here are some additional algebraic identities that are useful:
- (a2 - b2 = (a - b)(a + b))
- ((a + b)2 = a2 + 2ab + b2
- (a2 + b2 = (a + b)2 - 2ab
- ((a - b)2 = a2 - 2ab + b2
- ((a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- ((a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ca
- (a3 - b3 = (a - b)(a2 + ab + b2)
- (a3 + b3 = (a + b)(a2 - ab + b2)
- ((a + b)3 = a3 + 3a2b + 3ab2 + b3
- ((a + b)3 = a3 + b3 + 3ab(a + b)
- ((a - b)3 = a3 - 3a2b + 3ab2 - b3 = a3 - b3 - 3ab(a - b)
These identities are very helpful in simplifying and solving algebraic expressions.