What is the Identity a^3-b^3?

S6-SA1-0049

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The identity a^3 - b^3 is a special algebraic formula used to factorise (break down) expressions where one perfect cube is subtracted from another perfect cube. It helps simplify complex expressions into a product of two simpler factors. This identity is a fundamental tool for solving many problems in algebra.

Simple Example

Quick Example

Imagine you have two cube-shaped boxes. One box has a side length of 'a' and the other has a side length of 'b'. The volume of the first box is a^3, and the volume of the second is b^3. If you want to find the difference in their volumes, a^3 - b^3, this identity helps you express that difference in a simpler, factored form. For example, if a=5 and b=3, then 5^3 - 3^3 = 125 - 27 = 98. Using the identity, it would be (5-3)(5^2 + 5*3 + 3^2) = (2)(25 + 15 + 9) = (2)(49) = 98.

Worked Example

Step-by-Step

Factorise the expression 8x^3 - 27y^3 using the identity a^3 - b^3.

1. Identify the terms as perfect cubes: 8x^3 can be written as (2x)^3, and 27y^3 can be written as (3y)^3.
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2. Compare with the identity a^3 - b^3. Here, a = 2x and b = 3y.
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3. Recall the identity: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
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4. Substitute the values of 'a' and 'b' into the identity.
(2x - 3y)((2x)^2 + (2x)(3y) + (3y)^2)
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5. Simplify the terms inside the second bracket.
(2x - 3y)(4x^2 + 6xy + 9y^2)
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6. The factored form of 8x^3 - 27y^3 is (2x - 3y)(4x^2 + 6xy + 9y^2).

Answer: (2x - 3y)(4x^2 + 6xy + 9y^2)

Why It Matters

Understanding a^3 - b^3 is crucial for simplifying complex equations in engineering, physics, and even AI/ML algorithms. Engineers use it to design structures efficiently, while data scientists might use similar algebraic principles to optimise calculations. It's a foundational concept that supports advanced problem-solving in many exciting careers.

Common Mistakes

MISTAKE: Writing (a-b)^3 instead of (a-b)(a^2 + ab + b^2) | CORRECTION: Remember that (a-b)^3 expands to a^3 - 3a^2b + 3ab^2 - b^3, which is different from a^3 - b^3. The identity is a specific factoring pattern.

MISTAKE: Forgetting the '+' sign in the middle term of the second bracket (a^2 - ab + b^2) | CORRECTION: For a^3 - b^3, the second bracket is (a^2 + ab + b^2). The sign of 'ab' is positive. Only for a^3 + b^3 is it negative.

MISTAKE: Incorrectly squaring terms like (2x)^2 as 2x^2 | CORRECTION: When squaring a term like (2x), both the number and the variable get squared, so (2x)^2 = 2^2 * x^2 = 4x^2.

Practice Questions

Try It Yourself

QUESTION: Factorise x^3 - 8 | ANSWER: (x - 2)(x^2 + 2x + 4)

QUESTION: Factorise 125p^3 - 64q^3 | ANSWER: (5p - 4q)(25p^2 + 20pq + 16q^2)

QUESTION: If (x-y) = 3 and (x^2 + xy + y^2) = 15, find the value of x^3 - y^3. | ANSWER: 45

MCQ

Quick Quiz

Which of the following is the correct identity for a^3 - b^3?

(a - b)(a^2 - ab + b^2)

(a - b)(a^2 + ab + b^2)

(a + b)(a^2 + ab + b^2)

(a - b)^3

The Correct Answer Is:

Option B is the correct identity. Option A has a negative 'ab' term, which is incorrect for a^3 - b^3. Option C is for a^3 + b^3. Option D is the expansion of a binomial cubed, not the factored form of the difference of cubes.

Real World Connection

In the Real World

In computer graphics, when designing 3D models for games or animation, complex shapes are often broken down into simpler geometric forms. Identities like a^3 - b^3 can be used in the background algorithms to efficiently calculate volumes or differences between volumes of objects, helping create realistic virtual worlds like those in popular Indian games or educational apps.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of its variables. | FACTORISE: To express a polynomial as a product of two or more polynomials. | PERFECT CUBE: A number or expression that is the result of cubing an integer or another expression (e.g., 8 is 2^3, x^3 is x cubed). | BINOMIAL: An algebraic expression of two terms.

What's Next

What to Learn Next

Great job mastering a^3 - b^3! Now, you should explore the identity for 'What is a^3 + b^3?'. It's very similar but with small, important changes, and understanding both will make you a pro at factoring cubic expressions!