What is (a+b)^3 Identity? | Simple Explanation for Class 10

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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 the cube of a sum of two terms, 'a' and 'b'. It means multiplying (a+b) by itself three times. This identity simplifies calculations and is expressed as a^3 + b^3 + 3ab(a+b) or a^3 + b^3 + 3a^2b + 3ab^2.

Simple Example

Quick Example

Imagine you have two friends, 'A' and 'B', who each scored some runs in a cricket match. If you want to find the 'cube' of their combined score, (A+B)^3, this identity helps you break it down easily. Instead of doing (A+B) x (A+B) x (A+B), you can directly use the formula to get the total in terms of individual scores cubed and their products.

Worked Example

Step-by-Step

Let's expand (2x + 3)^3 using the identity (a+b)^3 = a^3 + b^3 + 3ab(a+b).
---Step 1: Identify 'a' and 'b'. Here, a = 2x and b = 3.
---Step 2: Substitute 'a' and 'b' into the formula: (2x)^3 + (3)^3 + 3(2x)(3)(2x + 3).
---Step 3: Calculate the cubes: (2x)^3 = 8x^3 and (3)^3 = 27.
---Step 4: Calculate the middle term: 3(2x)(3) = 18x. So, the expression becomes 8x^3 + 27 + 18x(2x + 3).
---Step 5: Distribute 18x into (2x + 3): 18x * 2x = 36x^2 and 18x * 3 = 54x.
---Step 6: Combine all terms: 8x^3 + 27 + 36x^2 + 54x.
---Step 7: Arrange in standard form (highest power first): 8x^3 + 36x^2 + 54x + 27.
Answer: The expansion of (2x + 3)^3 is 8x^3 + 36x^2 + 54x + 27.

Why It Matters

Understanding algebraic identities like (a+b)^3 is crucial for solving complex problems in various fields. Engineers use these principles to design structures and calculate volumes, while data scientists in AI/ML might use similar polynomial expansions for modeling data. These skills are foundational for careers in engineering, physics, and even medicine, where understanding growth patterns can involve cubic relationships.

Common Mistakes

MISTAKE: Students often forget to cube the coefficient when 'a' or 'b' is a term like 2x, writing 2x^3 instead of (2x)^3. | CORRECTION: Remember to cube the entire term, including its numerical coefficient. (2x)^3 = 2^3 * x^3 = 8x^3.

MISTAKE: Students sometimes write only a^3 + b^3 and forget the 3ab(a+b) part. | CORRECTION: The identity is not just the sum of cubes; it also includes the product terms. Always remember the full formula: a^3 + b^3 + 3ab(a+b).

MISTAKE: Incorrectly distributing 3ab into (a+b), especially with negative signs or complex terms. | CORRECTION: Be careful with distribution. 3ab(a+b) means 3ab multiplied by 'a' AND 3ab multiplied by 'b', resulting in 3a^2b + 3ab^2.

Practice Questions

Try It Yourself

QUESTION: Expand (p + 4)^3. | ANSWER: p^3 + 12p^2 + 48p + 64

QUESTION: Expand (3m + 1)^3. | ANSWER: 27m^3 + 27m^2 + 9m + 1

QUESTION: If x + y = 5 and xy = 6, find the value of x^3 + y^3 using the identity. (Hint: (x+y)^3 = x^3 + y^3 + 3xy(x+y)) | ANSWER: 35

MCQ

Quick Quiz

Which of the following is the correct expansion of (a+b)^3?

a^3 + b^3

a^3 + b^3 + 3ab

a^3 + b^3 + 3a^2b + 3ab^2

a^3 + b^3 + a^2b + ab^2

The Correct Answer Is:

Option C correctly represents the full expansion of (a+b)^3, which is a^3 + b^3 + 3a^2b + 3ab^2. Options A, B, and D are incomplete or incorrect expansions of the identity.

Real World Connection

In the Real World

Imagine a town planner calculating the volume of water tanks in a new housing society. If the side length of a cubical tank is (x+2) meters, they would use (x+2)^3 to find its volume. Similarly, in computer graphics, when designing 3D models of objects, these cubic expansions help in calculating complex shapes and volumes for realistic rendering.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all values of the variables involved. | EXPANSION: The process of multiplying out the terms in an expression. | CUBE: The result of multiplying a number or expression by itself three times. | COEFFICIENT: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression. | POLYNOMIAL: An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

What's Next

What to Learn Next

Great job understanding (a+b)^3! Next, you should explore the identity (a-b)^3. It's very similar but involves negative signs, which will test your understanding of algebraic operations and prepare you for even more complex polynomial manipulations.