Question

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.



1) -2, -1, 1

2) i, 4

3) i, 2, -√3

4) 1, 4, 1 + √2

Answer 1

first one is easiest. start with
\[p(x)=(x+2)(x+1)(x-1)\] and multiply out

Answer 2

you should get
\[p(x)=x^3+2 x^2-x-2\] for the first one
second one you need to know that if
\[i\] is a root then so is
\[-i\] so again multiply
\[p(x)=(x-i)(x+i)(x-4)=(x^2+1)(x-4)\]

Answer 3

then they get more complicated. third one you need
\[p(x)=(x+i)(x-i)(x-2)(x+\sqrt{3})(x-\sqrt{3})\]
\[=x^5-2 x^4-2 x^3+4 x^2-3 x+6\]

Answer 4

last one is...
\[p(x)=(x-1)(x-4)(x-(1+\sqrt{2})(x-(1-\sqrt{2}))\]