To really find a solution, Positively

During my research in Engineering, I came across this problem.

There is a complex valued function, with the domain as the imaginary axis.

\[P(j\omega) = P_R(\omega^2)+j\omega P_I(\omega^2)\]

I need to find the value of $P(j\omega)$ , at the sequence of points where it is real and draw appropriate conclusions.

There are two scenarios here. (1) Both $P_R(x)$ and $P_I(x)$ are polynomials in $x$ . (2) Not case 1.

For case 1 : As we are interested in values of $P_R(\omega^2$ ) at those $\omega$ , where $P_I(\omega^2)=0$ , it is obvious that we are interested ONLY in positive and real solutions to $P_I(x)=0$ .

My question to the community is this.

Are there any methods to find the real roots (or better positive real roots) of a polynomial, without actually finding all the roots?*

If there is no simpler solution, than finding all the roots; then the procedure which I envision would become more computationally complex than a conventional technique that is more than 80 years old.

If case 1 is worth pursuing, then case 2 could be an interesting extension.

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$