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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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