a3 b3 Formula- Proof, Solution and Examples

Q: How do you prove a3 +b3?

Proof: We take, a3+b3+3ab(a+b)=(a+b)2 (a+b) => a3+b3 = (a+b)2 (a+b) - 3ab(a+b) = (a+b) {(a+b)2-3ab} = (a+b)(a2+b2+2ab-3ab) => a3+b3 = (a+b)(a2+-ab+b2)

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Algebraic Equation- For (a3 b3 Formula)

An algebraic equation, often known as a polynomial equation, is a mathematical equation of the form:

P = 0

P is a polynomial having coefficients in a field, most commonly the field of rational numbers. Many authors solely use the word algebraic equation to describe univariate equations, or polynomial equations with only one variable. A polynomial equation, on the other hand, can have several variables. Polynomial equation is frequently favoured over algebraic equation when there are multiple variables (multivariate scenario).

a³ b³ Formula Proofs:

(a + b)³ = a³ + b³ + 3ab (a + b)
(a – b)³ = a³ – b³ – 3ab (a-b)
a³ + b³ = (a + b) (a² – ab + b²)
a³ – b³ = (a – b) (a² + ab + b²)
a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ca)
a + b = (a³ + b³ ) / (a² – ab + b²)

a3 b3 Formula: Solutions And Examples

Problem: Factorize 27x³ – 64y³
Solution: We can write 27x³ – 64y³ as (3x)³ – (4y)³.

Therefore, by using the formula:

a³ – b³ = (a – b) (a² + ab + b²),

we get the factors as:

(3x – 4y) (9x² + 16y² + 12xy)

Problem: Factorize x³ + 8

Solution: The given equation x³ + 8 can be written as:

x³ + 2³. therefore, it is of the form,

a³ + b³.

Hence, by using the formula:

a³ + b³ = (a + b) (a² – ab + b²)

we get the factors as:

(x + 2) (x² – x . 2 + 2²)

= (x + 2) ( x² – 2x +4)

Problem: Find 8³ – 3³
Solution: By using the formula:

a³ – b³ = (a – b) (a² + ab + b²),

we get the solution as:

(5) × (8² + 9 + 24) = 485

Problem: Factorize (2x + y)³ – (x + 2y)³
Solution: As this given equation is of the form a³ – b³,

Therefore, by using the formula:

a³ – b³ = (a – b) (a² + ab + b²),

we get the factors as:

(2x + y – x – 2y) [(2x + y)² + (x + 2y)² + (2x + y) (x + 2y)]
= (x – y) (7x² + 7y² + 13xy)

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a3 b3 Formula: FAQs

Ques: How do you prove a3 +b3?

Ans: Proof: We take, a3+b3+3ab(a+b)=(a+b)2 (a+b)

=> a3+b3 = (a+b)2 (a+b) – 3ab(a+b)

= (a+b) {(a+b)2-3ab}

= (a+b)(a2+b2+2ab-3ab)

=> a3+b3 = (a+b)(a2+-ab+b2)

Ques: What is this algebra?

Ans: Algebra is a field of mathematics that aids in the depiction of problems and situations using mathematical expressions. To construct a meaningful mathematical statement, it uses variables like x, y, and z, as well as mathematical operations like addition, subtraction, multiplication, and division.

Ques: What is the Algebra formula?

Ans: Algebra Formulas: What Are They? An algebraic formula is a mathematical or algebraic rule represented as an equation. It’s a two-sided equation with algebraic expressions on both sides. The algebraic formula is a quick and easy way to answer difficult algebraic problems.