Let $P(x)=x^2+bx+c$, where $b$ and $c$ are integers.

algebra-precalculuspolynomialsquadraticsquarticsroots

Let $P(x)=x^2+bx+c$, where $b$ and $c$ are integers. If $P(x)$ is a
factor of both $f(x)=x^4+6x^2+25$ and $g(x)=3x^4+4x^2+28x+5$, then

  1. $P(x)=0$ has imaginary roots
  2. $P(x)=0$ has roots of opposite sign
  3. $P(1)=4$
  4. $P(1)=6$

My Attempt:

$f(x)=x^4+6x^2+25$ is always positive, i.e. no roots.

This implies that, $P(x)=0$ has no real roots either.

$\therefore$ Option (1) is correct.

I have no idea how to calculate Option (3) or (4)

All I could see was $f(1)=32$, $g(1)=40$, hinting me at (3).

Any hints would be really helpful thanks.

Best Answer

The quartic $f(x)$ has $4$ roots: $1\pm2i$ and $-1\pm2i$. Of these, only the first $2$ are roots of $g(x)$. Therefore, $P(x)=(x-1-2i)(x-1+2i)=x^2-2x+5$. Now, note that $P(1)=4$.

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