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Question 1

If $$x^2 + x = 1$$, then the value of $$\frac{x^7 + 34}{x + 2}$$ is equal to

Solution

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$$x^2 = 1 - x$$

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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