A minus B the whole cube formula

A-B Whole Cube

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

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