Pre-Calculus Math

Question

I need help with this question? Please? In part a it says to find the zeros of p, real and complex including repetitions?

A polynomial P is given. P(x) = x^4+ 4×^2 + 4

(a) Find all zeros of P, real and complex. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

(b) Factor P completely.

Answer 1

x^4 +4x^2+ 4

= (x^2+2)^2

=(x^2+2)(x^2+2)

maybe the easiest way to see this is let x^2=z

then P(z) = z^2+4z +4 which factors into (z+2)^2

then substitute back to get P(x)= (x^2+2)^2

x^2+2=0

x^2=-2

x= +/-isqr2

zeros are isqr2, -isqr2 each with multiplicity 2

or

x= isqr2, isqr2, -isqr2, -isqr2

there are no real zeros, just 4 imaginary zeros

which also means

if you factor further you only get imaginary irrational factors

P(x)= (x-isqr2)^2(x+isqr2)^2

=(x+isqr2)(x+isqr2)(x-isqr2)(x-isqr2)

usually you stop with the integer rational factors

as the "complete" factors