Solve the reciprocal equation $x^4-2x^3-x^2-2x+1=0$.

Solve the reciprocal equation x⁴-2x³-x²-2x+1=0.

Exercise 4(d) · Q6

x^4-2x^3-x^2-2x+1=0

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Step 1. The coefficients $1,-2,-1,-2,1$ read the same forwards and backwards, so this is a class-one reciprocal equation of even degree $4$ .

Step 2. Since $x=0$ is clearly not a root, divide throughout by $x^2$ :

x^2-2x-1-\frac2x+\frac1{x^2}=0.

Step 3. Group the symmetric pairs: $\Big(x^2+\dfrac1{x^2}\Big)-2\Big(x+\dfrac1x\Big)-1=0$ .

Step 4. Let $t=x+\dfrac1x$ , so $x^2+\dfrac1{x^2}=t^2-2$ . The equation becomes

(t^2-2)-2t-1=0\ \Longrightarrow\ t^2-2t-3=0\ \Longrightarrow\ (t-3)(t+1)=0,

so $t=3$ or $t=-1$ . …