Question

Determine the number of positive real roots and imaginary roots for equation 3 $x^7$ + 2 $x^5$ + 4 $x^3$ + 11x - 12 = 0.

• A. 1, 6
• B. 2, 5
• C. 3, 4
• D. 4, 3

Answer 1

The correct option is A

1, 6

Solution: If f(x) = 3 $x^7$ + 2 $x^5$ + 4 $x^3$ + 11x - 12 = 0

+ + + + -

It has only one sign change.

f(x) = 0 has exactly one +ve roots

f(-x) = -3 $x^7$ - 2 $x^5$ - 4 $x^3$ - 11x - 12 = 0

- - - - -

There is no change of sign in f(-x). So, number of negative real roots = 0

Also, f(0) = - 12≠ 0, 0 is not the root of the f(x) = 0

f(x) has only one real root which is positive

Number of imaginary roots
= Total number of roots - Total no. of real roots

= 7 - 1 = 6