Polynomial roots!

Algebra Level 3

How many distinct real roots does the polynomial

$P(x)=9x^6-90x^5+229x^4-96x^3+857x^2-3370x+4925$

have?

2

5

4

3

0

6

1

1 solution

Otto Bretscher
Dec 27, 2018

We can write $P(x)=(x-5)^2\left(9x^4+4(x-7)^2+1\right)$ , so that there is only $\boxed{1}$ distinct real root, $x=5$ .

Since the other four solutions consist of two complex conjugate pairs, I suppose it's fair to say that 5 as a double-root is equivalent to the complex conjugate pair (5,-0) (5,+0). In my childhood, I always thought that the concept of double (or triple, etc.) roots was silly, but I suppose these can actually hint at a more general structure (complex numbers).

Steven Chase - 2 years, 5 months ago

Wow, I liked how you rearrange the terms for the quartic factor so effortlessly. I tried Descartes' rule of signs at $P(x) = (x-5)^2 ( 9x^4 + 4x^2 - 56x + 197 )$ , but made no progress. So I resorted to c̶h̶e̶a̶t̶i̶n̶g̶ graphing, which I'm not satisfied with.

Pi Han Goh - 2 years, 5 months ago

"Rearranging the terms of the quartic factor" is just a matter of completing the (right) square; we practiced that in high school "ad nauseam."

Otto Bretscher - 2 years, 5 months ago

Yeah my bad. I usually don't like to see a bracket inside another bracket.

Pi Han Goh - 2 years, 5 months ago

Polynomial roots!

1 solution

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