A-B Whole Cube
Table of Contents
A minus B the whole cube formula is an algebraic identity which is used to find the cube of binomial. The expression of this A minus B Whole cube formula is (a – b)3 = a3 – 3ab(a-b) – b3 or (a – b)3 = a3 – 3a2b + 3ab2 – b3
- The a minus b whole cube is equal to a cubed minus b cubed minus 3 a b times a minus b.
How to derive A Minus B the whole cube
Lets understand this (A-B)3 Formula in details. this formula is the cube of the difference of two variables or terms. Here we will see the other expressions of A minus B the whole cube formula or you can say other forms of (A-B)3 formula.
- To find the formula of (a – b)3, we will just multiply (a – b) (a – b) (a – b).
- (a – b)3 = (a – b)(a – b)(a – b)
- = (a2 – 2ab + b2)(a – b)
- = a3 – a2b – 2a2b + 2ab2 + ab2 – b3
- = a3 – 3a2b + 3ab2 – b3
- = a3 – 3ab(a-b) – b3
- Therefore, (a – b)3 formula is: (a – b)3 = a3 – 3a2b + 3ab2 – b3
List of Some important Algebraic formula
- (a – b)3 = a3 – b3 – 3ab (a – b)
- (a + b)3 = a3 + b3 + 3ab (a + b)
- ( a − b ) 3 = a 3 − b 3 − 3 a b ( a − b )
- ( a − b ) 3 = a 3 − b 3 − 3 a b ( a − b )
- ⟹ ( a − b ) 3 = a 3 − b 3 − 3 a 2 b + 3 a b 2.
- ⟹ ( a − b ) 3 = a 3 − b 3 − 3 a b ( a − b )
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- (a + b) (a – b) = a2 – b2
- (x + a) (x + b) = x2 + (a + b) x + ab
- (x + a) (x – b) = x2 + (a – b) x – ab
- (x – a) (x + b) = x2 + (b – a) x – ab
- (x – a) (x – b) = x2 – (a + b) x + ab
- (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
- (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
- (x + y + z)2 = x2 + y2 + z2 + 2xy +2yz + 2xz
- (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
- (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
- (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
- x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz -xz)
- x2 + y2 = 1212 [(x + y)2 + (x – y)2]
- (x + a) (x + b) (x + c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc
- x3 + y3 = (x + y) (x2 – xy + y2)
- x3 – y3 = (x – y) (x2 + xy + y2)
- x2 + y2 + z2 – xy – yz – zx = 1212 [(x – y)2 + (y – z)2 + (z – x)2]
Solved examples using A-B Whole Cube formula
Now you can understand this (A-B)3 formula in detail by solving some good examples
Example 1: Solve the following expression using suitable algebraic identity: (2p – 3q)3
Solution: Using (a – b)3 Formula, (a – b)3 = a3 – 3a2b + 3ab2 – b3 Here in given question, we can assume a = 2p and b = 3q, now put these variables in formula
= (2p)3 – 3 × (2p)2 × 3q + 3 × (2p) × (3q)2 – (3q)3
= 8p3 – 36p2q + 54pq2 – 27q3
Example 2: Find the value of p3 – q3 if p – q = 5 and pq = 2 using (a – b)3 formula.
Solution: We have to find: p3 – q3
In the question, given terms are : p – q = 5 and pq = 2 . Now to solve this question we will use (a – b)3 A minus B the whole cube formula, (a – b)3 = a3 – 3a2b + 3ab2 – b3 In this question Here, a = p; b = q
Therefore,
(p – q)3 = p3 – 3 × p2 × q + 3 × p × q2 – q3
(p – q)3 = p3 – 3p2q + 3pq2 – q3
53 = p3 – 3pq(p – q) – q3
125 = p3 – 3 × 2 × 5 – q3
p3 – q3 = 95
Example 3: Solve the following expression using (a – b)3 formula: (5p – 2q)3
Solution: We have to find: (5p – 2q)3 Using (a – b)3 Formula, (a – b)3 = a3 – 3a2b + 3ab2 – b3 in this question, a = 5p and b = 2q
= (5p)3 – 3 × (5p)2 × 2q + 3 × (5p) × (2q)2 – (2q)3
= 125p3 – 150p2q + 60pq2 – 8q3
FAQs on A Minus B the whole cube formula
(a – b)3 = a3 – 3a2b + 3ab2 – b3 or a3 – 3ab(a-b) – b3
(a – b)3 = a3 – 3a2b + 3ab2 – b3 or a3 – 3ab(a-b) – b3
(a – b)3 = a3 – 3a2b + 3ab2 – b3 or a3 – 3ab(a-b) – b3
(a – b)3 = a3 – 3a2b + 3ab2 – b3 or a3 – 3ab(a-b) – b3
(a – b)3 = a3 – 3a2b + 3ab2 – b3 or a3 – 3ab(a-b) – b3 The a minus b whole cube is equal to a cubed minus b cubed minus 3 a b times a minus b.
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