a3 b3 Formula- Proof, Solution and Examples

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.

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