Question

A real polynomial `P(x)` is the product of real linear and real quadratic factors. Can you that if a real quadratic factor has one zero, `a+bi`, then the other zero must be the complex conjugate, `a-bi`?

Hint: Let the quadratic be `alphax^2 + betax + gamma` and use the properties of complex conjugates

Answer 1

See below.

Explanation:

`P(a+ib)="Re"(P(a+ib))+i "Im"(P(a+ib))`

where `"Re"(cdot)`is the real component of `P(a+ib)` and
`"Im"(cdot)` is the real part of the imaginary component. Now

`P(a+ib)=0->"Re"(P(a+ib))=0` and `"Im"(P(a+ib))=0`

but

`"Re"(z) = (z+bar z)/2` where `bar z` is the conjugate of `z` and
`"Im"(z)=(z-bar z)/(2i)` so now if `P` has real coefficients, `P(x) = bar P(x)` and `bar(P(x))=P(bar x)` and also

`"Re"(P(a+ib)) = (P(a+ib)+P(a-ib))/2`

and

`"Im"(P(a+ib)) = (P(a+ib)-P(a-ib))/(2i)`

So if `P(a+ib)=0` then

`{(P(a+ib)+P(a-ib)=0),(P(a+ib)-P(a-ib)=0):}`

concluding `P(a-ib)=0`

Another version.

If `P(a+ib)=0` then `P(x)=Q_1(x)(x-(a+ib))` but `P(x)` is a real polynomial. That means all its coefficients real. So `P(x)` must have another root `x_0` such that `(x-x_0)(x-(a+ib))=x^2+alphax+beta` with `alpha, beta` real. Or

`x^2-(x_0+a+ib)x+x_0(a+ib) = x^2+alpha x+beta`

or

`{(-(x_0+a+ib)=alpha),(x_0(a+ib)=beta):}`

The feasible solution is

`x_0=a-ib`

because

`-(a-ib+a+ib) = -2a = alpha` and also

`(a-ib)(a+ib)=a^2+b^2=beta`

with both `alpha, beta` real.

So if `P(x)` is a real polynomial and `P(a+ib)=0` then necessarily `P(a-ib)=0` and `P(x)=Q_2(x)(x^2+alpha x + beta)` with `alpha = -2a` and `beta = a^2+b^2`

Answer 2

Use the quadratic formula...

Explanation:

If one of the real quadratic factors is:

`alphax^2+betax+gamma`

then using the quadratic formula, it has zeros:

`(-beta+-sqrt(beta^2-4alphagamma))/(2alpha)`

If `beta^2-4alphagamma >= 0` then the square root is real and the zeros are real, resulting in real linear factors.

Otherwise, `beta^2-4alphagamma < 0` so:

`sqrt(beta^2-4alphagamma) = (sqrt(4alphagamma-beta^2))i`

resulting in a complex conjugate pair of non-real zeros.