# Position of Roots of a Polynomial Equation

## Position of Roots of a Polynomial Equation

Some results on roots of an equation: The following are results on the roots of a polynomial equation with rational coefficients:

• An equation of degree n has n roots, real or imaginary.
• Imaginary roots always occur in pairs i.e., if 2 – 3i is a root of an equation, then 2 + 3i s also its root. Similarly, if 2 + √3 is a root of a given equation, then 2 – √3  is also its roots.
• An odd degree equation has at least one real root, whose sign is opposite to that coefficient to that of its last term, provided that the coefficient of highest degree term is positive.
• Every equation of an even degree whose constant term is negative and the coefficient of highest degree term is positive, has at least two real roots, one positive and one negative.

Position of Roots of a Polynomial Equation: If f(x) = 0 is an equation and a, b are two real numbers such that f(a) f(b) < 0, then the f(x) = 0  has at least one real root or an odd number of real roots between a and b. in case f(a)  and f(b)  are of the same sign, then either no real root or an even number or real roots of f(x) = 0 lie between a and b.

Deductions:

1. Every equation of an odd degree has at least one real root, whose sign is opposite to that of its last term, provided that the coefficient of first tern is positive.
2. Every equation of an even degree last term is negative and the coefficient of first term positive, has at least two real roots, one positive and one negative.
3. If an equation has only one change of sign, it has one positive Root and no more.
4. If all the terms of an equation are positive and the equation involves no odd powers of x, then all its roots are complex.

Example: If a, b, c, d ϵ R such that a < b < c < d, then show that the roots of the equation (x -a) (x – c) + 2(x – b) (x – d) = 0 are real and distinct.

Solution: Let f(x) = (x – a) (x – c) + 2(x – b) (x – d)

Then,

f (a) = 2(a – b) (a – d) > 0   [∵ a – b < 0 and a – d < 0]

f (b) = (b – a) (b – c) < 0     [∵ b – a > 0 and b – c < 0]

f (d) = (d – a) (d – c) > 0     [∵ d – a > 0 and d – c > 0]

So, a root of f(x) = 0 lies between a and b are real and distinct.