A polynomial f x with real coefficients and leading coeffi

A polynomial f ( x ) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f ( x ) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.

A polynomial f( x) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f( x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R. 4+1i,-9+ degree 4

Solution

zeros: x = 4 +i

There would be another complex conjugate root: x = 4 -i

x = -9 +i

There would be another complex conjugate root: x = -9 -i

Leading coefficient is 1

So, f(x) = 1*(x - 4 -i)(x -4 +i)(x +9 -i)(x +9 +i)

=(x^2-4x +ix -4x -4i -ix +16 +4i +1)(x^2 +9x +ix+9x+ 81+9i -ix -9i +1)

=(x^2 -8x +17)(x^2 +18x+82)