Question 1
If $$x^2 + x = 1$$, then the value of $$\frac{x^7 + 34}{x + 2}$$ is equal to
Solution
From $$x^2 = 1 - x$$ the higher powers can be reduced step by step, giving $$x^3 = 2x - 1$$, $$x^4 = 2 - 3x$$, $$x^5 = 5x - 3$$, $$x^6 = 5 - 8x$$ and $$x^7 = 13x - 8$$. Hence $$\frac{x^7 + 34}{x + 2} = \frac{13x - 8 + 34}{x + 2} = \frac{13(x + 2)}{x + 2} = 13$$.
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