Formula of a plus b whole cube and a minus b whole cube
The Correct Answer and Explanation is:
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Correct Answer:
- (a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3(a+b)3=a3+3a2b+3ab2+b3
- (a−b)3=a3−3a2b+3ab2−b3(a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3(a−b)3=a3−3a2b+3ab2−b3
Explanation
The expressions (a+b)3(a + b)^3(a+b)3 and (a−b)3(a – b)^3(a−b)3 are examples of binomial cubes. These represent the cube of a binomial — an algebraic expression that has two terms. Cubing a binomial means multiplying the expression by itself three times.
To expand (a+b)3(a + b)^3(a+b)3, the expression becomes:(a+b)(a+b)(a+b)(a + b)(a + b)(a + b)(a+b)(a+b)(a+b)
Using the distributive property (also called the FOIL method for two binomials), the first two binomials multiply to form:(a+b)(a+b)=a2+2ab+b2(a + b)(a + b) = a^2 + 2ab + b^2(a+b)(a+b)=a2+2ab+b2
Multiplying this result by (a+b)(a + b)(a+b) again:(a2+2ab+b2)(a+b)=a3+3a2b+3ab2+b3(a^2 + 2ab + b^2)(a + b) = a^3 + 3a^2b + 3ab^2 + b^3(a2+2ab+b2)(a+b)=a3+3a2b+3ab2+b3
This result gives the expanded form of (a+b)3(a + b)^3(a+b)3.
Similarly, for (a−b)3(a – b)^3(a−b)3, the binomial cube expands as:(a−b)(a−b)(a−b)(a – b)(a – b)(a – b)(a−b)(a−b)(a−b)
The first part becomes:(a−b)(a−b)=a2−2ab+b2(a – b)(a – b) = a^2 – 2ab + b^2(a−b)(a−b)=a2−2ab+b2
Multiplying by (a−b)(a – b)(a−b) gives:(a2−2ab+b2)(a−b)=a3−3a2b+3ab2−b3(a^2 – 2ab + b^2)(a – b) = a^3 – 3a^2b + 3ab^2 – b^3(a2−2ab+b2)(a−b)=a3−3a2b+3ab2−b3
This is the expanded form of (a−b)3(a – b)^3(a−b)3.
Both formulas follow a clear pattern:
- The first and last terms are the cubes of the original binomial components.
- The middle terms involve combinations of a2ba^2ba2b and ab2ab^2ab2, scaled by a factor of 3.
- The signs alternate depending on whether the original expression involves addition or subtraction.
These identities are fundamental in algebra and are often used in factoring, polynomial expansion, and solving equations.
