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:
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 !!