# Introduction

Taylor

Nicholas Chuquet (1445-1500) invented a method for solving polynomials iteratively.

Daniel Bernoulli (1700-1782) expressed the largest root of a polynomial as the limit of the ratio of the successive power sums of the roots.

Leonard Euler (1707-1783) tried to find solutions of polynomial equations of degree n as sums of nth roots, but failed.

Polynomial is a special type of function or a special type of algebraic expression.

Consider p(x) = 4x^{3} - 3x^{2} + 4x - 5. This is a function of x.

If we substitute any real value a for x, we get a real number p(a). For example

p(1) = 4(1)^{3} - 3(1)^{2} + 4(1) - 5 = 0

p(2) = 4(2)^{3} - 3(2)^{2} + 4(2) - 5 = 23

p(- 2) = 4(- 2)^{3} - 3(- 2)^{2} + 4(- 2) - 5 = - 57 and so on.

So p(x) is a function whose domain is the set of all real numbers. Here we see that only non-negative powers of x are involved in each term and the coefficient of each power of x is a real number. Any such function is called a polynomial over the real numbers.

The word polynomial means an algebraic expression consisting of many terms (poly = many) involving powers of the variable.

# Definition of a Polynomial

A function p(x) of the form p(x) = a_{0}+ a

_{1}x + a

_{2}x

^{2}+ â€¦ a

_{n}x

^{n}where a

_{0}, a

_{1}, a

_{2}, â€¦ , a

_{n}are real numbers and n is a non-negative integer, is called a polynomial in x over reals.

The real numbers a

_{0}, a

_{1}, a

_{2, }â€¦, a

_{n}are called the coefficients of the polynomial.

If a

_{0}, a

_{1}, a

_{2}, â€¦, a

_{n}are all integers, we will call it polynomial over integers.

If co-efficients are rational numbers, we call it a polynomial over rationals. For examples:

6x

^{2}- 12x + 7 is a polynomial over integers

is a polynomial over rationals and x^{2} + - 2 is a polynomial over reals.

The variables may be x, y, z, a, b, c etc.

**Example :** 3y^{2} + 2y - 3 (or) 3a^{2} - 3a + 5

Factor theorem is a special case of remainder theorem. It is used to solve equations of higher power; find the remainder when one polynomial is divided by another; and factorise equations.

**Louis Lagrange**

When solving equations, up to the fourth power, we use formulae. But, for equations of fifth power and above, there is no analytical solution. This was proved by Louis Lagrange using group theory. One of the ways to solve such equations is by applying the concept of remainder and factor theorem.

**Remainder Theorem**

Let p(x) be any polynomial of degree â‰¥ 1, and a be any real number. If p(x) is divided by (x - a), then the remainder will be p(a).

**Proof**

Consider that p(x) is divided by x - a and we get quotient q(x) and remainder r(x). So we have,

p(x) = (x - a) q(x) + r(x)

where r(x) = 0 or degree r(x) â‰¤ 0 degree (x - a)

Since degree of (x - a) is 1, either r(x)=0 or

degree of r(x)=0 (<1). So r(x) is a constant, say r. Hence for all values of x,

p(x) = (x - a) q(x) + r where r is a constant.

In particular, for x = a,

p(a) = 0 [ q(a)] + r= r

**Factoring a Polynomial**

With reference to the properties of addition, multiplication, subtraction and division, polynomials behave exactly as integers do. Every polynomial can be expressed as the product of linear polynomials or polynomial of degree less than that of the given polynomial and which do not have any other factors. Here, we will study the methods of expressing a polynomial as the product of polynomials which cannot be further factorized. This method is called factorization of polynomials.

# Types of Polynomials

**Linear Polynomial**

A polynomial of degree one is called a linear polynomial. The general form of a linear polynomial is ax + b where a and b are real numbers and a â‰ 0

**Example :**

(a) 2x + 3

(b) x - 5,

(c) x + 2 etc.

**Quadratic Polynomial**

A polynomial of degree two is called quadratic polynomial. The general form of quadratic polynomial is ax^{2} + bx + c where a, b and c are real numbers and a â‰ 0

**Example :**

(a) 2x^{2} + 3*x* + 4

(b) 5x^{2} - 3x + 1

(c) 7x^{2} + 8

(d) x^{2} + 7x + etc.

**Cubic Polynomial**

A polynomial of degree three is called cubic polynomial.

**Example :**

(a) x^{3} - 2x^{2} + 3x - 5

(b) x^{3} + y^{3}

(c) x^{3} + 1

(d) 2y^{3} + 2y + 1 etc.

**Biquadratic Polynomial**

A polynomial of degree 4 is called biquadratic polynomial.

**Example :**

(a) 5x^{4} + 2x^{3} + 8x^{2} + 11x + 12

(b) x^{4} + 5

(c) 2x^{4} + x + 1 etc.

Polynomial of degree 4 is also called quadratic polynomial.

**Some Important Formulae**

(A + B)^{2} = A^{2} + B^{2} + 2AB

(A - B)^{2} = A^{2} + B^{2} - 2AB

(A + B)^{2} = (A - B)^{2} + 4AB

(A - B)^{2} = (A + B)^{2} - 4AB

A^{2} - B^{2 }= (A + B) (A - B)

(A + B)^{3} = A^{3} + B^{3} + 3AB(A + B)

(A - B)^{3} = A^{3} - B^{3} - 3AB(A - B)

A^{3} + B^{3} = (A + B) (A^{2} - AB + B^{2})

A^{3} - B^{3} = (A - B) (A^{2} + AB + B^{2})

A^{4} - B^{4} = (A^{2} + B^{2}) (A + B) (A - B)