When a question asks for a symmetric expression, finding each possible value of the variable can introduce radicals that later cancel. An identity may determine the requested expression directly from the information given.
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Keep the nonzero condition visible
In this original exercise, x is real and nonzero, and x + 1/x = 3. Find x3 + 1/x3.
Solving x2 - 3x + 1 = 0 is possible, but the requested sum treats x and its reciprocal symmetrically. First ask whether the given sum can be transformed into the target.
Square with the cross term included
Squaring the given equation gives x2 + 2 + 1/x2 = 9, because x(1/x) = 1. Therefore x2 + 1/x2 = 7.
The answer is not nine: that would omit the cross term. Stating why the product is one also identifies precisely where the nonzero condition is used.
Multiply and collect the middle terms
Multiply the quadratic sum by the original sum. The left side expands to x3 + x + 1/x + 1/x3, while the right side is 7 × 3 = 21.
The two middle terms add to three. Subtracting them leaves x3 + 1/x3 = 18. No individual root has been selected, so the result applies to every x satisfying the original condition.
Generalise only after checking the first steps
If Sn = xn + x-n, then S0 = 2 and S1 = 3. Multiplication gives Sn+1 = 3Sn - Sn-1 for integers n ≥ 1. Thus S2 = 7 and S3 = 18 agree with the direct work.
This is an original AMC 12 practice problem, not an official question. The habit transfers to more extended algebra practice, including preparation for AIME: identify the target structure, justify the identity and check conditions before doing extra algebra.
Continue reading: AMC 12 Sequences: Derive a Geometric Sum by Subtracting Two Lines.
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Official information and further reading: Source 1
