Question

Please help answer these questions for chapter polynomials:-

1. Let p, q and r be such that p + q=r and pqr = 30. What is the value of p³ + q³ - r³
2. Factorise : 9 p² - (p²-4)²

Answer 1

For the first question, the value of p³ + q³ - r³ is -3pq(p + q). For the second question, the factorised form of 9p² - (p² - 4)² is -1(p⁴ - 17p² + 16).

Step-by-step explanation:

1. Let's solve the first question step-by-step:

We are given that p + q = r and pqr = 30. We need to find the value of p³ + q³ - r³.

1. Since p + q = r, we can substitute r in terms of p and q to get p³ + q³ - (p + q)³.
2. Expanding (p + q)³ gives us p³ + 3p²q + 3pq² + q³.
3. Substituting the values from the previous step and rearranging the terms, we get p³ + q³ - (p³ + 3p²q + 3pq² + q³).
4. Now, simplifying the expression gives us p³ + q³ - p³ - 3p²q - 3pq² - q³.
5. Combining like terms, we have -3p²q - 3pq².
6. Finally, factoring out a common factor of -3pq gives us -3pq(p + q).

Therefore, the value of p³ + q³ - r³ is -3pq(p + q).

2. To factorise 9p² - (p² - 4)², let's simplify step-by-step:

1. Expanding (p² - 4)² gives us p⁴ - 8p² + 16.
2. Now, substituting the value from step 1, we have 9p² - (p⁴ - 8p² + 16).
3. Combining like terms, we get 9p² - p⁴ + 8p² - 16.
4. Rearranging the terms, we have -p⁴ + 17p² - 16.
5. Finally, factoring out a common factor of -1, we get -1(p⁴ - 17p² + 16).

Therefore, the factorised form of 9p² - (p² - 4)² is -1(p⁴ - 17p² + 16).