Question

help please!! (show work) A polynomial f(x) with a real coefficient and leading coefficient 1 has the given zeros and degree. Express f(x) as a product of linear and quadratic polynomials with real coefficient that are irreducible over R. -1,0,-2-i; degree 4

Answer 1

@hartnn would you mind helping me

Answer 2

one of the roots is -2-i

what could be the other root?

Hint : if a+bi is one root, then a-bi is also one root for real co-efficient polynomials

Answer 3

So would the other root be -2+i

Answer 4

yes! correct.

So now we have all 4 roots.
-1,0, -2-2i, -2+i

when 'a' is a root, (x-a) is a factor.
so, when -1 is a root, (x-(-1)) =(x+1) is a factor!

can you similarly get another 3 factors??

just multiply them together to get your f(x)! :)

Answer 5

x (x+2+i) (x+2-i) would be the other factors right

Answer 6

exactly!
now you can multiply (x+2+i) (x+2-i) out to get a quadratic equation :)

Answer 7

So when I multiply (x+2+i)(x+2-i) I got x^2 +6x+5

Answer 8

I added it wrong it was x^2 +4x+5

Answer 9

yes,x^2 +4x+5 is correct.

so your f(x) is x(x+1)(x^2+4x+5)
thats it!