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If $$x^2 + x = 1$$, then the value of $$\frac{x^7 + 34}{x + 2}$$ is equal to
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If $$x^2 + x = 1$$, then the value of $$\frac{x^7 + 34}{x + 2}$$ is equal to
$$x^2 = 1 - x$$
We can find
$$x^3 = x - x^2 = x - (1-x) = 2x-1$$
$$x^4 = 2x^2 - x = 2(1-x) -x =2-3x $$,
$$x^5 = 2x-3x^2 = 2x-3(1-x) = 5x - 3$$,
$$x^6 = 5x^2-3x = 5(1-x)-3x = 5 - 8x$$
$$x^7 = 5x-8x^2 = 5x-8(1-x) = 13x - 8$$.
Hence $$\dfrac{x^7 + 34}{x + 2} = \dfrac{13x - 8 + 34}{x + 2} = \dfrac{13(x + 2)}{x + 2} = 13$$.
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