Question

For the given equation x${}^9$ + 5x${}^8$ - x${}^3$ + 7x+3=0 how many maximum real roots are possible?

• A. 3
• B. 5
• C. 6
• D. 7

Answer 1

The correct option is B

5

Let f(x) =x${}^9$ + 5x${}^8$ - x${}^3$ + 7x+3 = 0

We can solve this question easily using Descartes' Rule

According to Descartes' rule maximum number of positive real roots = number of sign

changes in f(x) = 2

Similarly, Maximum number of negative real roots = number of sign changes in f(-x)

Note: To find f(-x) replace "x” by "-x” in each instance

therefore f(-x) = - x${}^9$ + 5x${}^8$ + x${}^3$ - 7x+3

Maximum number of negative real roots = number of sign changes in f(-x) = 3

Zero cannot be a root because constant part is also involved in equation.

So maximum number of real roots = 2 + 3 = 5


DESCARTES' RULE (Points to Remember)
Maximum number of positive real roots = number of sign changes in f(x)
Maximum number of negative real roots = number of sign changes in f(-x)
If a constant term is present, do not consider 0 as a root or else, you need to consider it