Question

If the zeros of the polynomial x³ − 3x² + x + 1 are a −d, a and a + d, then a + d is

(a) a natural number
(b) an integer
(c) a rational number
(d) an irrational number

Answer 1

$$\begin{gathered} \left(\text{d}\right)\mathrm{an}\mathrm{irrational}\mathrm{number} \\ \text{Since}a-d,a\mathrm{and}a+d\text{are the zeroes of}{x}^3-3x^2+x+1,\text{we have:} \\ \text{Sum of zeroes=3} \\ \text{=>}a-d+a+a+d=3 \\ =>3a=3 \\ =>a=1 \\ \text{So, the zeroes are}(1-d),1\mathrm{and}(1+d). \\ \mathrm{Now},\text{product of the zeroes}=-1 \\ =>(1-d)\times 1\times (1+d)=-1 \\ =>1-d^2=-1 \\ =>d^2=2 \\ =>d=\sqrt{2} \\ \therefore a+d=(1\pm \sqrt{2}),\text{which is irrational}\text{.} \end{gathered}$$