If x2 - √8 x + 1 = 0, then x3 + x-3 =
2020
If x2 - √8 x + 1 = 0, then x3 + x-3 =
- A.
5√8
- B.
2√8
- C.
4√8
- D.
8√8
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Correct answer: A
CONCEPT: For a non-zero x, a quadratic relation can be divided by x to obtain a reciprocal sum. The cubic reciprocal identity is x3 + x-3 = (x + x-1)3 - 3(x + x-1).
APPLICATION: Use the given quadratic to find the reciprocal sum, then substitute that sum in the cubic identity.
Since x2 - √8 x + 1 = 0 and x is non-zero, divide by x: x + x-1 = √8.
Apply the identity: x3 + x-3 = (x + x-1)3 - 3(x + x-1).
Substitute x + x-1 = √8: (√8)3 - 3√8 = 8√8 - 3√8 = 5√8.
CROSS-CHECK: Let an = xn + x-n. Then a0 = 2, a1 = √8, a2 = (√8)2 - 2 = 6, and a3 = √8·6 - √8 = 5√8.
Result: The value of x3 + x-3 is 5√8.