What is the formula for a3-b3? Know all about this algebraic formula

Algebra can be described as a mathematical term that assists people in interpreting complex situations by using mathematical equations involving variables like x, Z, and y. We will present to readers the algebraic formula for A3-B3 and its proof derivative as well as some examples. This formula is more significant not only for high school students but also for those studying for competitive exams like NTSE, NDA, AFCAT, SSC, Railways, and so on.

a3-b3 Formula

The formula of a3-b3 in algebra is a³-b³= (a-b) (a²+ab+b²). It is also known as a cube minus b cube formula. You can use the a3-b3 formula in maths to solve different algebraic problems.

a³-b³ = (a – b) (a²+ab+b²)

Basics of Algebraic Equation

It is the Algebraic Equation in the subject of mathematics is that two algebraic equations are at a level equal to each other. The combination of variables constants, and coefficients make what is known as the Algebraic Equation.

The Algebraic Equation always provides the equal formula for unknown constants, variables and coefficients. Because every side share the same value, it’s classified as a balanced equation. It is shown in the form P=0, where P is the polynomial.

a3 b3 Formula Chart

There are many other formulas related to a³ b³ in algebra. Let’s have a look at the a³ b³ Formula Chart.

a3-b3 Formula Proof

There are various algebraic formulas in mathematics to be used for solving the equation. Here the complete proof derivation of the a3-b3 formula is given below.

Since we know the formula of (a-b)³= a³-b³-3ab(a-b)

a³-b³= (a-b)³+ 3ab(a-b)

a³-b³= (a-b) [(a-b)² + 3ab]

After the expansion of the equation on Right Hand Side (RHS),

a3-b³= (a – b) (a²+b²-2ab+3ab)

a3-b³= (a – b) (a²+b²+ab)

a3-b³= (a – b) (a²+ab+b²)

The above equation can be proved by the following steps:

First considering the Right Hand Side (RHS) of the equation,

(a-b) (a²+ ab+ b²)= a(a² + ab + b²) – b(a² + ab + b²)

(a-b) (a²+ ab+ b²)= (a – b)(a² + ab + b²) = a3 + a2b + ab2 – a2b – ab2 – b3

Taking out the common factor (a-b),

(a-b) (a2+ ab+ b2)= a3 + a2b – a2b + ab2– ab2 – b3

Bringing the similar terms together and canceling out the probable terms such as a2b and ab2,

(a-b) (a2+ ab+ b2)= a3-b3 (RHS) which is equal to Left Hand Side (LHS). Hence this formula is proved.

In this way, you should remember the a³-b³ formula to start your calculation in the examination.

a³-b³ = (a – b) (a²+ab+b²)

a³-b³ Formula Examples with Solutions

By memorizing this formula, you can easily calculate the mathematical problems.

Question 1: Factorize 125a³-27b³?

Solution: As we know that the expression of 125a³-27b³ can be written as (5a)³-(3b)³

By employing the a³-b³ formula,

a³-b³ = (a – b) (a²+ab+b²)

Now, we will get the factors of the given equation as,

(5a)³-(3b)³ = (5a-3b) (25a² + 15ab + 9b²)

Here 5a indicates a and 3b indicates b of the well-proven a³-b³ formula.

Question 2: Factorize (3a + b)³ – (2a + b)³?

Solution: As we know that the given equation is in the form of the a3-b3 formula,

a³ – b³= (3a + b)³ – (2a + b)³

Here a is indicated by (3a + b) while b is indicated by (2a + b)

By using the formula of a³-b³,

a3-b3= (a – b) (a2+ab+b2)

(3a + b)³ – (2a + b)³  = (3a + b – 2a – b) [(3a+b)² + (3a + b)(2a + b) +(2a + b)²]

(3a + b)³ – (2a + b)³  = (3a + b – 2a – b) [(9a² + b² + 6ab) + (6a² +3ab +2ab +b²) +(4a² + b2 +4ab)]

(3a + b)³ – (2a + b)³  = (3a + b – 2a – b) (9a² +6a² +4a² +b² +b² +b² +6ab +3ab +2ab +4ab)

(3a + b)³ – (2a + b)³  = (3a + b – 2a – b) (19a² + 3b² + 15ab)

Hence, one factor of the given equation will be (3a + b – 2a – b) and the other factor will be (19a2 + 3b2 + 15ab).

Question 3: Find out the 5³-2³?

Solution: As we know that the given equation is in the form of the a³-b³ formula,

a³ – b³= (a – b) (a2+ab+b2)

Here a is indicated by 5 while b is indicated by 2

So, 5³-2³ = (5-2) (25 + 10 + 4)

5³-2³ = 3 (39) = 117 will be the answer.

a3-b3 Formula- FAQ’s

Q1. What do you mean by Algebra Formula?

Ans. An Algebra Formula is a well-recognized mathematical rule that is expressed in the form of an equation. It shows the algebraic expressions on both sides. It is easier to solve difficult mathematical problems.

Q2. What is the a3-b3 formula?

Ans. For easier calculation, you should remember the a3-b3 formula as given a³-b³= (a – b) (a²+ab+b²).

Q3. What is Algebra?

Ans. Algebra is the sphere of mathematics that represents the situations and problems in mathematical expressions by using operations like addition, and subtraction. etc.

Q4. What is the a3+b3 formula?

The formula a3+b3 is a³ + b³ = (a + b) (a² – ab + b²).