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Maths-

General

Easy

Question

For every x ε R, the polynomial x⁸ - x⁵ + x² - x + 1 is :

positive
never positive
positive as well as negative
negative

hint Hint:

In the given question, we are required to find the value of the polynomial $x to the power of 8 minus x to the power of 5 plus x squared minus x plus 1$ .

The correct answer is:
positive

Step by step solution:
Let f(x) = $x to the power of 8 minus x to the power of 5 plus x squared minus x plus 1$
$rightwards double arrow$ f(x) = $x to the power of 5 left parenthesis x cubed minus 1 right parenthesis plus x left parenthesis x minus 1 right parenthesis plus 1$
$rightwards double arrow$ f(x) = $x to the power of 5 space left parenthesis space x minus 1 right parenthesis space left parenthesis x squared plus x plus 1 right parenthesis$ $plus x left parenthesis x minus 1 right parenthesis space plus 1$
$rightwards double arrow$ f(x) = $x left parenthesis x minus 1 right parenthesis$ $left curly bracket x to the power of 4 left parenthesis x squared plus x plus 1 right parenthesis plus 1 right curly bracket plus 1$
$x to the power of 4 greater or equal than 0 space$ and $x squared plus x plus 1 greater or equal than 0$
CASE 1 : If x does not belong to (0,1)
Then, f(x) $greater than 0$ [ since $x left parenthesis x minus 1 right parenthesis$ $greater or equal than 0$ ]
CASE 2: If $x space element of space left parenthesis 0 comma 1 right parenthesis$ , then f(x) > 0
Thus, the polynomial is positive for all x $element of$ R.
Hence, option(c) is the correct option.

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