Form the monic cubic equation with rational coefficients, two of whose roots are $2+\sqrt3$ and $2-\sqrt3$, the third root being $1$.

Form the monic cubic equation with rational coefficients, two of whose roots are 2+√3 and 2-√3, the third root

Exercise 4(a) · Q8

2+\sqrt3

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Step 1. The conjugate pair $2+\sqrt3,\ 2-\sqrt3$ combines to the rational quadratic factor

\big(x-(2+\sqrt3)\big)\big(x-(2-\sqrt3)\big)=(x-2)^2-3=x^2-4x+4-3=x^2-4x+1.

Step 2. The required cubic is $(x^2-4x+1)(x-1)=0$ .

Step 3. Expand:

$(x^2-4x+1)(x-1)=x^3-4x^2+x-x^2+4x-1=x^3-5x^2+5x-1.$ …