Show that the polynomial x^8 -x^7+x^2 -x+15=0 Has no real roots.

Need this for an exam Urgently any kind of help would be appreciated

No vote yet

2 votes

×

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

TopNewestLet K = x^8-x^7 + x^2 - x = x(x-1)(x^6+1)

Observe that x must be in (0,1) for K to be negative

x^6+1 takes values in (1,2) for x in the range (0,1)

The minimum value of (x)(x-1) is -1/4

Thus K >= -1/2

Log in to reply

From the above, K + 15 >= 29/2

Thus x^8 - x^7 + x^2 - x + 15 is always positive and thus has no real roots.

Log in to reply

descartes rule of signs only must a limit on the maximum real roots.i dont think thats the only idea to be applied here.

Log in to reply

I do not know about how to prove it has no real roots. You can show that it has no rational roots.

Log in to reply

actually if we are given with a polynomial where f(x) = 0 ., then for f(x) to have no real roots, the condition is that f(-x) should not have ny change of signs...... so in the given polynomial f(-x) = (-x)^8 -(-x)^7 +(-x)^2 -(-x)+15 => f(-x) = x^8 +x^7+x^2+x+15

so thr is no change of signs in f(-x)....and hence there r no real roots , all of the roots r imaginary...

here i've used the descartes rule of signs

Log in to reply

there can be 4 or 2 and maybe 0 positive real roots for the polynomial according to descarte's theorem.

Log in to reply

But that is only concluded for negative roots, what about real positive roots?

Log in to reply

no thts not correct actually......the maximum number of negative real roots of a polynomial equation f(x)=0 is the number of change of signs from positive to negative and negative to positive in f(-x)

and the number of positive real roots of a polynomial equation f(x)=0 is the number of change of signs from positive to negative and negative to positive in f(x)

and if f(-x) has got no sign change, then the polynomial equation has complex roots..

Log in to reply

Log in to reply

so wht can u conclude from this?.... i guess thr is a flaw in the question

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

so u've finished ur 12th this yr?...r u a cbse student?

Log in to reply

Log in to reply

Log in to reply

Comment deleted Jul 14, 2013

Log in to reply

Log in to reply

Decartes' rule of signs? Maybe you could use that.

Log in to reply