IIT JEE , AIEEE Forum - Mathematics , Algebra

trisha

show that the equation x⁴+4x³-4x-13=0 has two real roots only and find the position of roots.

irlmaks

by rule of sighns it is obvi since it has a root in [-4,-3]

it has another root in [1,2]

shine

hey can ne one tell whts this descartes rule of signs
plzzzz explain ............

ankur.kkhurana

ya what is descartes rule of sign

trisha

mr.irlmaks , since 1,2,-3,-4 are smaller numbers we can locate two roots by trial and error method.But my question is that the other two roots may also be real and they may be in some intervel like (-33325226,-48888893) and (-2543412321,-9475836752) i.e., how can we say that other two roots are not real?

magiclko

descartes rule is a method of determining the maximum number of positive and negative real roots of a polynomial.

For positive roots, start with the sign of the coefficient of the lowest (or highest) power. Count the number of sign changes n as you proceed from the lowest to the highest power (ignoring powers which do not appear). Then n is the maximum number of positive roots. Furthermore, the number of allowable roots is n, n-2, n-4 .... For example, consider the polynomial

Since there are three sign changes, there are a maximum of three possible positive roots.

For negative roots, starting with a polynomial f(x), write a new polynomial f(-x) with the signs of all odd powers reversed, while leaving the signs of the even powers unchanged. Then proceed as before to count the number of sign changes n. Then n is the maximum number of negative roots. For example, consider the polynomial

and compute the new polynomial

In this example, there are four sign changes, so there are a maximum of four negative roots.

irlmaks

trisha

thats flithy

see by rule of signs it has a max. of one +ve and one -ve root

existance can be checked by differentiation

meathods like increasing..... etc

trisha

mr.irlmaks , plz keenly observe there are three sign changes in f(-x)

that implies atmost 3 negative roots.how can u say that ther is only one negative root?

magiclko or any expert plz solve this for me..

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