Question 22 of 43intermediate⚖️ EvaluateEssay3 marks

According to you, who is correct and why? Try to combine more such identities and find new results. (Context: James and Reshma were talking about algebraic identities they learnt in school. James: (a - b)^2 (a + b) = (a2a^{2} - 2ab + b2b^{2})(a + b). Reshma: I have a different idea. (a - b)^2 (a + b) = (a - b) [(a - b) (a + b)] = (a - b)(a2a^{2} - b2b^{2}). I will find this product to get the answer.)

Correct Answer

Both James and Reshma are correct. James expanded (a−b)2(a - b)^2 using the identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 to get (a2−2ab+b2)(a+b)(a^2 - 2ab + b^2)(a + b). Reshma grouped the terms and applied the identity (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2 to get (a−b)(a2−b2)(a - b)(a^2 - b^2). Both methods are valid and yield the same final product.

We can combine more such identities to find new results. For instance, combining (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 with (a+b)(a + b) gives the identity for the cube of a binomial: (a+b)3=(a+b)(a2+2ab+b2)=a3+3a2b+3ab2+b3(a + b)^3 = (a + b)(a^2 + 2ab + b^2) = a^3 + 3a^2b + 3ab^2 + b^3. Similarly, combining (a+b)2(a + b)^2 and (a−b)(a - b) yields (a+b)2(a−b)=(a+b)(a2−b2)=a3+a2b−ab2−b3(a + b)^2(a - b) = (a + b)(a^2 - b^2) = a^3 + a^2b - ab^2 - b^3.

Exercise: Think and Reflect | Q: 1 | (Page 15)
For More Understanding

Explanation

The textbook context highlights that algebraic identities can be manipulated and combined in different ways. James uses the expansion of a squared binomial, while Reshma uses the difference of squares identity. Both are mathematically sound. The context also introduces finding new identities by combining existing ones, explicitly showing the derivation of (a+b)3(a+b)^3.

Solution Steps

  1. Step 1: Verify James's approach by applying the identity (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2.

  2. Step 2: Verify Reshma's approach by applying the identity (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2.

  3. Step 3: Conclude both are correct as both are valid applications of algebraic identities.

  4. Step 4: Combine identities like (a+b)2(a+b)^2 and (a+b)(a+b) to find new results such as (a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.

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