# Polynomial

If n be a positive integer, the expression:

F(x) = a_{0}x^{n} + a_{1}x^{n - 1}+â€¦â€¦.+ a_{n}

Where all the coefficients a_{0}, a_{1}, a_{2,}â€¦..a_{n} are constants and a_{0 }â‰ 0 is called a 'polynomial of degree n in x'. When a polynomial of degree n is equated to zero, we get an algebraic equation of degree n. The general nth degree equation will be taken as f(x) â‰¡ a_{0} x^{n} + a_{1}x^{n - 1} + a_{2}x^{n - 2} + â€¦â€¦â€¦+ a_{n} = 0

An equation of the second degree is called a quadratic equation, of the third degree a cubic equation and the fourth degree a bi-quadratic or quartic equation.

**Division Algorithm****:** If f(x) and g(x) be polynomials of degree n and m (< n) respectively, then there exists a polynomial (x) of degree (n - m) and a polynomial r(x), which is either identically zero or is of degree less than m, such that f(x) = g(x) (x) + r(x)

Here, (x) is called the 'quotient' and r(x) the remainder.

**Factor Theorem****:** If is a root of the equation f(x) = 0, then the polynomial f(x) is divisible by (x - ) without a remainder.

**Remainder Theorem****:** The remainder, when the polynomial f(x) is divided by x - , is f().

**Fundamental Theorem of Algebra****:** If f(x) is a polynomial in x, then the equation f(x) = 0 has a root, real or imaginary.

From here it can be concluded that *every equation of nth degree has n and only n roots.*

**Theorem** **I:** In an equation with real coefficients, complex roots occur in conjugate pairs.

**Theorem II**: In an equation with rational coefficients, irrational roots of the type a + âˆšb occur in conjugate pairs.

**Theorem III**: Every rational root of the equation

x^{n} + a_{1}x^{n - 1} +â€¦.a_{n} = 0, where each a_{1}(i = 1, 2, â€¦â€¦.., n) is an integer, must be an integer. Moreover, every such root must be a divisor of the constant a_{n}.

** **

**Common roots to two equations**: In order to find the common roots of the two equations f(x) = 0 and g(x) = 0, find the H.C.F. of the two polynomials f(x) and g(x) and equate this H.C.F. to zero. The roots of this equation are the required common roots.

**Multiple roots**: In order to determine the equal or repeated roots of an equation f(x) = 0, find the common roots of the equations f(x) = 0 and f'(x) = 0. Therefore, equate to zero the H.C.F. of f(x) and f'(x) and the roots of equation so obtained are the repeated roots of equation f(x) = 0.

# Relations between roots and coefficients

Let f(x) â‰¡ a_{0}x^{n} + a_{1}x^{n - 1 }+ a_{2}x^{n - 2} + â€¦â€¦..+ a_{n - 1}x + a_{n} = 0

be the general equation of the nth degree and let _{1,}_{2},â€¦â€¦â€¦, _{n}, be its roots. Then we have

f(x) â‰¡ a_{0}x^{n} + a_{1}x^{n - 1} + a_{2}x^{n - 2} â€¦â€¦..+a_{n}

â‰¡ a_{0} (x - _{1})(x - _{2}) (x - _{3}) â€¦..(x - _{n})

Equating coefficients of like powers of x on both sides, we have

âˆ‘_{1} = - a_{1}/a_{0}

âˆ‘_{1}_{2} = a_{2}/a_{0}

âˆ‘_{1} _{2}_{3} = - a_{3}/a_{0 }

**Elementary transformations:**

Some elementary transformations are:

1. To transform an equation into another equation with roots equal in magnitude but opposite in sign to the roots of the given equation, change x into - x in the given equation.

2. To transform the given equation into another, whose roots are m times the roots of the given equation, multiply its second coefficient by m, third by m^{2}, fourth by m^{3} and so on, and the last by m^{n}.

3. To transform an equation into another whose roots are reciprocals of the roots of the given equation, change x into 1/x in the given equation and then multiply by x^{n}.

4. To transform an equation into another whose roots are the roots of the given equation diminished by h, put y = x - h, i.e. x = y + h in the given equation. This can be done by dividing f (x) by (x - h) and each resulting quotient by (x - h), until the last quotient is a constant, the successive quotients that are left in the process of division being the coefficients.

5. The transform an equation into another in which a particular term is removed, first decrease the roots of the given equation by, say h, and then choose h in such a way that the coefficient of the unwanted term vanishes. With this value of h, the transformed equation will have its particular term missing.

# Symmetric functions of the roots

A function in which all the roots of an equation are involved alike, so that its value remains unaltered when any two of the roots are interchanged, is called a 'symmetric function' of the roots of the equation. The expressions + + ,^{2}+

^{2}+

^{2}+

^{2}+

^{3}+

^{2}, etc., are symmetric functions of the three roots , , of a cubic.