Question

The value of the polynomial x power 8 - x power 5 + x square - x + 1
(a) positive for all the real numbers
(b) negative for all the real numbers
(c) 0
(d) depends on value of x

Answer 1

Answer:

Step-by-step explanation:

By factorisation of polynomial p(x)=

we get:

p(x)= {x}^{5} ( {x}^{3} −1)+x(x−1)+1=x(x−1)( {x}^{4} ( {x}^{4} +x+1)+1)+1

 

We know that, x 4 ≥0 and x 2 +x+1>0 ∀ x∈R.

Case I : x  ∈(0,1)

As x(x−1)≥0,

⇒p(x)≥1>0 ...... (positive in range)

Case II : [x∈(0,1)]

Maximum value of [−x(x−1))] is 1/4 at x=1/2.

And maximum value of [x

4 (x^2 +x+1)+1] is 4 at x=1 as its derivative is greater than zero in the range.

∴x(x−1)(x

4 (x ^2 +x+1)+1)>−(1/4)×4=(−1) ...... [as maxima of the terms are not coincident]

⇒p(x)>0, x∈(0,1)

∴p(x)>0∀ x∈ R

HOPE IT HELPS U BROTHER