Analyzing a fourth degree polynomial

Question

Let $a,b$ and $c$ be real numbers. Then prove that the fourth degree polynomial in $x$
$acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac$ has either 4 real roots or 4 complex roots.
I have never solved a fourth degree polynomial and don't know the conditions for it to have real/complex roots. How do we approach this problem?

Answer 1

Notice that the given expression,

$$acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac = 0$$

can be factorized into (see below for derivation):

$$(ax^2 + bx + c)(cx^2 + bx + a) = 0$$

The discriminant for each is the same, $b^2 - 4ac$.

If this common discriminant is zero or more, then the roots for both $ax^2 + bx + c = 0$ and $cx^2 + bx + a = 0$ are all real (possible with multiplicity $2$). If not, they are all imaginary. Hence the roots are either all real or all imaginary.

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Here's how I did the factorization:

$$acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac$$ $$=acx^4 + abx^3 + bcx^3 + a^2x^2 + b^2x^2 + c^2x^2 + abx + bcx + ac$$ $$=(acx^4 + abx^3 + a^2x^2) + (bcx^3 + b^2x^2 + abx) + (c^2x^2 + bcx + ac)$$ $$=ax^2(cx^2 + bx + a) + bx(cx^2 + bx + a) + c(cx^2 + bx + a)$$ $$=(ax^2 + bx + c)(cx^2 + bx + a)$$