Let's apply Polya’s four-step problem-solving process in this case.

STEP 1: Understand the problem

Suppose that the cubic equation f(x) = ax³ + bx² + cx+ d = 0 has roots α, β, γ.

f(x) = ax³ + bx² + cx+ d = 0

x = α, β, γ

Find a new cubic equation g(x) with the roots h(α); h(β); h(γ).

Given: x = h(α); h(β); h(γ)

Find: g(x) = 0

STEP 2: Develop a plan to solve the problem

Remainder and Factor Theorems tell us that: f(α) = f(β) = f(γ) = 0

Since g(x) = 0, we can apply the law of conservation of energy and we pick: g(x) = f(γ) = 0

Both sides must contain x. And the link between x and γ is x = h(γ). Manipulate it: γ = h^-1(x)

STEP 3: Carry out plan

g(x) = f(γ) = 0

g(x) = f[h^-1(x)] = 0

g(x) = px³ + qx² + rx+ s = 0

STEP 4: Look back and check the answer.

The roots of the cubic equation px³ + qx² + rx+ s = 0 should be exactly the same as h(α); h(β); h(γ).

By the way, this sorcery is called “by means of a substitution”, and the substitution is γ = h^-1(x) = 2x / (x − 1). Although this problem is inherently A-level / STPM level, it can be solved using SPM Form 4 Add Math Functions knowledge + Standard Algebraic Manipulations (SAM). Thanks ystiang for he has done a good job!

Your approach is basically correct. The mistake probably occurred at the Numerator of the Sum: 1 − z^(N − 1). When you add up the terms up to the last term n in a converging geometric series, the Sum formula should compute the sum of n+1 terms (compare the LHS and the RHS of the equation below).



Another mistake occurred at the denominator part when you multiplied two complex conjugates to produce a real number. Mathematicians expand to proper polar forms when doing the complex multiplication. The best practice is to avoid writing shorthands of polar form r∠θ when doing arithmetic operations, even though the work is tedious.

To find out how to solve the rest of the problem, please check out the Examples 23 & 24 (found in Parts 11 & 12 of my Complex Number tutorial note). I believe they would be helpful, because your problem is exactly derivable from the fundamental examples.

As a smart student, choose the selective question fastidiously, because complex trigonometric proof often requires one to go the extra mile and then return. For example, in proving the cos part (Real), the student inevitably has to expand the sine part (imaginary) as well, even though it is not required to be proven. For solving a 10-mark question, you need the 20-mark worth time. One thing good is that most A-level/STPM complex problems cannot run away far from the fundamental examples.

i have a question here...              refer to part b, why can't we consider the moving of charge form infinity to C is driven by external force? accoring to the question the work done is either done or by the electric field...

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