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What is the Identity (a-b)^3?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity (a-b)^3 is a special algebraic formula that helps us expand or simplify expressions where a difference of two terms is raised to the power of three. It means multiplying (a-b) by itself three times. This identity helps us solve problems faster without doing lengthy multiplications.
Simple Example
Quick Example
Imagine you have a square plot of land with side 'a' meters. You decide to reduce one side by 'b' meters for a small pathway. If you wanted to build a cube-shaped water tank with the new side (a-b), its volume would be (a-b)^3. The formula helps you find this volume directly.
Worked Example
Step-by-Step
Let's expand (5-2)^3 using the identity.
Step 1: Identify 'a' and 'b'. Here, a = 5 and b = 2.
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Step 2: Recall the identity: (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
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Step 3: Substitute the values of 'a' and 'b' into the identity: (5)^3 - 3(5)^2(2) + 3(5)(2)^2 - (2)^3.
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Step 4: Calculate each term: 5^3 = 125.
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Step 5: Calculate the second term: -3 * (25) * 2 = -150.
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Step 6: Calculate the third term: 3 * 5 * (4) = 60.
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Step 7: Calculate the fourth term: -2^3 = -8.
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Step 8: Add all the terms: 125 - 150 + 60 - 8 = 27.
So, (5-2)^3 = 3^3 = 27. Both methods give the same answer.
Answer: 27
Why It Matters
Understanding algebraic identities like (a-b)^3 is crucial in many fields. In Engineering, it helps design structures and calculate volumes efficiently. In Physics, it's used in formulas for motion and energy. Even in AI/ML, complex algorithms often rely on simplifying polynomial expressions, saving computational time.
Common Mistakes
MISTAKE: Forgetting the alternating signs in the expansion, often writing a^3 + 3a^2b + 3ab^2 + b^3 | CORRECTION: Remember the pattern: + - + - for (a-b)^3. It's a^3 - 3a^2b + 3ab^2 - b^3.
MISTAKE: Incorrectly calculating 3a^2b or 3ab^2, especially squaring 'a' or 'b' last or forgetting to multiply by 3. For example, calculating 3a^2b as (3a)^2b | CORRECTION: Always square 'a' first (a^2), then multiply by 3 and 'b'. Same for 3ab^2: square 'b' first (b^2), then multiply by 3 and 'a'.
MISTAKE: Confusing (a-b)^3 with (a+b)^3 or simply expanding it as a^3 - b^3 | CORRECTION: (a-b)^3 is NOT a^3 - b^3. It's a complete expansion with four terms. The (a+b)^3 identity has all positive signs.
Practice Questions
Try It Yourself
QUESTION: Expand (x-3)^3. | ANSWER: x^3 - 9x^2 + 27x - 27
QUESTION: If a = 4 and b = 1, find the value of (a-b)^3 using the identity. | ANSWER: 27
QUESTION: Simplify the expression (2p-q)^3. | ANSWER: 8p^3 - 12p^2q + 6pq^2 - q^3
MCQ
Quick Quiz
Which of the following is the correct expansion of (2x-1)^3?
8x^3 - 12x^2 + 6x - 1
8x^3 + 12x^2 - 6x + 1
8x^3 - 1
2x^3 - 12x^2 + 6x - 1
The Correct Answer Is:
A
Option A is correct because when a=2x and b=1, substituting into a^3 - 3a^2b + 3ab^2 - b^3 gives (2x)^3 - 3(2x)^2(1) + 3(2x)(1)^2 - (1)^3, which simplifies to 8x^3 - 12x^2 + 6x - 1. Other options have incorrect signs or terms.
Real World Connection
In the Real World
Engineers designing parts for a satellite at ISRO might use such identities. If a critical component has a dimension 'a' and needs to be reduced by a tiny amount 'b' for a snug fit, and its volume is cubic, the (a-b)^3 identity helps calculate the new volume precisely. This ensures perfect fit and function in space applications.
Key Vocabulary
Key Terms
IDENTITY: An equation that is true for all possible values of its variables. | EXPANSION: Writing out a product of terms as a sum of terms. | CUBE: Raising a number or expression to the power of 3. | TERM: A single number or variable, or numbers and variables multiplied together.
What's Next
What to Learn Next
Great job understanding (a-b)^3! Next, you should explore the identities for (a+b+c)^2 and (a^3 + b^3) or (a^3 - b^3). These build on the basic cube identities and are frequently used in more complex algebraic problems and higher mathematics.
