Which of the following is true for the identity (a + b)³?

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Question: Which of the following is true for the identity (a + b)³?

Options:

Correct Answer: a³ + 3a²b + 3ab² + b³

Solution:

(a + b)³ = a³ + 3a²b + 3ab² + b³.

Which of the following is true for the identity (a + b)³?

Which of the following is true for the identity (a + b)³?

Step 1: Start with the expression (a + b)³.
Step 2: Recognize that (a + b)³ means (a + b) multiplied by itself three times: (a + b) × (a + b) × (a + b).
Step 3: First, calculate (a + b) × (a + b). This gives you a² + 2ab + b² (this is the square of a + b).
Step 4: Now, take the result from Step 3, which is (a² + 2ab + b²), and multiply it by (a + b) again.
Step 5: Distribute (a + b) across (a² + 2ab + b²). This means you will multiply a and b with each term in (a² + 2ab + b²).
Step 6: When you multiply a with each term, you get a³ + 2a²b + ab².
Step 7: When you multiply b with each term, you get ba² + 2bab + b³, which simplifies to b³ + 2ab².
Step 8: Now combine all the terms from Steps 6 and 7: a³ + 2a²b + ab² + b³ + 2ab².
Step 9: Combine like terms: a³ + 3a²b + 3ab² + b³.
Step 10: Conclude that (a + b)³ = a³ + 3a²b + 3ab² + b³ is the final identity.

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