For the following tutorial problems, it helps to know/remember/understand/apply the following identities (in addition to all other standard/famous identities you learn in high school maths):

By the way, I hope you also know how to derive the above.Let me mention two methods to derive the above :

Method I: Using polynomial division in three variable, divide the dividend by the divisor .

Method II: Assume that is a polynomial with roots a, b and c. So, we know by the fundamental theorem of algebra that . Now, we also know that a, b and c satisfy P(X). Now, proceed further and complete the proof.

Let us now work on the tutorial problems below:

1) If , prove that

2) If , prove that .

Prove the following identities:

3)

4)

5)

6)

7)

8)

9)

10)

11) Prove that

12) If3 , prove that

13) If , prove that

14) If , prove that

15) If , then prove that

16) If , then prove that

17) If , then prove that

18) Prove that

19) If prove that

20) If , , , find the value of

21) Prove that

22) Prove that

23) if , prove that .

24) If , , , prove that

25) If , prove that

26) If , simplify:

27) Prove that the equation is equivalent to the equation , hence show that the only possible values of x and y are: ,

28) If , prove that and therefore that and are the only possible solutions.

Good luck for the PreRMo August 2019 !!

Regards,

Nalin Pithwa