Answer:
[tex]x^{3}-(4+\sqrt{6})x^{2}+4\sqrt{6}x[/tex]
Step-by-step explanation:
The roots given are 0,4 and [tex]\sqrt{6}[/tex].
We are required to find the polynomial function with the least degree
having the roots as 0,4 and [tex]\sqrt{6}[/tex].The minimum number of roots it can have is three and hence the least degree polynomial should be a cubic function.Hence the polynomial is of the form:[tex]f(x)=ax^{3}+bx^{2}+cx+d[/tex]
where a=1 (given in the question)
sum of the roots = [tex]-\frac{b}{a}[/tex]=[tex]4+\sqrt{6}[/tex]
∴b=[tex]-(4+\sqrt{6})[/tex]
product of the roots taken two at a time = [tex]\frac{c}{a}=4\sqrt{6}[/tex]
∴c=4[tex]\sqrt{6}[/tex]
product of the roots = [tex]-\frac{d}{a}[/tex] = 0
∴d=0
Hence the polynomial function becomes [tex]x^{3}-(4+\sqrt{6})x^{2}+4\sqrt{6}x[/tex]
