write a polynomial function with least degree and a leading coefficient of 1 with given zeros: a. 5 and 3+2√7 b. 2, 3, 1-i

Question

write a polynomial function with least degree and a leading coefficient of 1 with given zeros:

a. 5 and 3+2√7

b. 2, 3, 1-i

Answer 1

You did not say the polynomial must have RATIONAL coefficients,

but I will assume you want REAL coefficients...

a)

(x-5)(x - (3 - 2rad7)) =

x^2 - 5x - (3-2rad7)x + 10rad7 =

x^2 - x( 5 + (3-2rad7)) + 10rad7 =

x^2 - (8-2rad7)x + 10 rad7

Now, if you want RATIONAL coefficients:

(x-5) [(x - 3)-2rad7] [(x-3)+2rad7] =

(x-5) [ (x-3)^2 - 4*7] =

(x-5) [ (x^2 -6x + 9 - 28]=

(x-5) [ x^2 - 6x - 19]

(b) (x-2)(x-3) [( x-1)-i][ (x-1)+i]=

(x-2)(x-3) [ (x-1)^2+1] =

(x-2)(x-3) [ x^2 -2x +2] =

(x^2 -5x+6)(x^2-2x+2) =