Here, by the word polynomial, i am referring to those polynomials, who have a square root in an easy algebraic expressions. Yet how to get the intuition, on exactly how to increase the polynomial, therefore that us can aspect it, and then uncover its square root? Or is there any type of other technique also? prefer $sqrtfracx^24+frac1x^2-frac1x+frac x2-frac34$, At first look, we acquire instinct to first take LCM of every and add them. Yet after that, the numerator becomes $x^4+2x^3-3x^2-4x+4$. Now, (I don"t desire the answer) exactly how to obtain that instinct on factoring this polynomial? Or is over there a method? Remark: Is there any kind of faster, non-rigorous way also, of recognize the square root?


*


*

Given that your polynomial$$x^4+2x^3-3x^2-4x+4$$has essence coefficients, and also the highest possible coefficient is one and the lowest four, if over there are any rational solutions, they have to be one of$$pm frac11,quad pm frac21,quad pmfrac41.$$Here, you need to run over various divisors of $4$ and $1$.

You are watching: Can a polynomial have a square root

You will certainly surely succeed v that in this case, since you will discover two double roots...


*

evaluate your polynomial in ~ $y-2$ and see what happens. This is to get rid of power 3 here. And also hopefully simplify things.

edit

let $f(x)=x^4+colorred2x^3-3x^2-4x+4$. Now evaluate in ~ $y-colorred2$, i.e. Plug $y-colorred2$ for $x$:

$$f(y-colorred2)=(y-colorred2)^4+2(y-colorred2)^3...=?$$


*

The example you offered is no a polynomial as there are negative powers associated (e.g. 1/x^2). I guess that you typical to uncover a polynomial which, if you take the square root, yields an additional polynomial.This is only possible if all the linear components are in even multiplicity, i.e. The polynomial has 2n worths for every root. If that is the case, you can take the square root as you described.

See more: Hatching Duck Eggs: Can A Chicken Hatch Duck Eggs : Can Chickens Hatch Ducks?

response come comment

Ok as you execute not desire the answer, this is what I would do:- discover the rootsAs a hint, x = 1 is a root. Use factor division on the polynomial to mitigate its degree to a cubic equation and continue to find the roots.- inspect that all roots come in even frequencies, if not, us cannot take a "nice" square root.


Highly energetic question. Knife 10 call (not count the combination bonus) in order to answer this question. The reputation need helps safeguard this concern from spam and non-answer activity.

Not the price you're spring for? Browse various other questions tagged polynomials radicals or asking your own question.


deserve to the square root of details quantities be turned into an additional square root plus one arbitrary continuous
site style / logo © 2021 ridge Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.10.21.40537


her privacy

By click “Accept all cookies”, you agree ridge Exchange have the right to store cookies on your machine and disclose info in accordance through our Cookie Policy.