Polynomial

If n be a positive integer, the expression:

F(x) = a0xn + a1xn - 1+â€¦â€¦.+ an

Where all the coefficients a0, a1, a2,â€¦..an are constants and a0 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‰¡ a0 xn + a1xn - 1 + a2xn - 2 + â€¦â€¦â€¦+ an = 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

xn + a1xn - 1 +â€¦.an = 0, where each a1(i = 1, 2, â€¦â€¦.., n) is an integer, must be an integer. Moreover, every such root must be a divisor of the constant an.

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‰¡ a0xn + a1xn - 1 + a2xn - 2 + â€¦â€¦..+ an - 1x + an = 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‰¡ a0xn + a1xn - 1 + a2xn - 2 â€¦â€¦..+an

â‰¡ a0 (x - 1)(x - 2) (x - 3) â€¦..(x - n)

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

âˆ‘1 = - a1/a0

âˆ‘12 = a2/a0

âˆ‘1 23 = - a3/a0

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 m2, fourth by m3 and so on, and the last by mn.

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 xn.

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.