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Introduction
An algebraic expression in which the variables have only non negative powers is called polynomial.
Examples: x2+3xy2+4x5: 3x4 +4xy.
Factorisation is expressing a given expression or number as a product of its factors.
Fundamental of Polynomial
Constants and variables
Symbols having fixed values are called constants. Examples: 5, π, ¾, 1.5.
Symbols which can be assigned with various numerical values are called variables or literals.
Examples : x, y, y 2.
In the formula for circumference of a circle, c=2π r 2 and π are constants and c and r are variables
Degree and Types of Polynomials
In case of a polynomial of one variable the highest power of the variable is called the degree of the polynomial.
3y-7y3 has degree 3 and <5x4-3x2 has degree 4.
In case of polynomials with more than one variable, the sum of powers of the variables taken up and the highest sum so obtained is called the degree of the polynomial.
7y2x+8y4x+9y3 has degree 5
Properties of Polynomials
Addition and subtraction of two polynomials mean combining like terms.
We can perform multiplication and division also
Factorization of Polynomials
You know that any polynomial of the form p(a) can also be written as
p(a) = g(a) x h(a) + R(a) it implies that Dividend = Quotient X Divisor + Remainder.
If the remainder is zero, then p(a) = g(a) x h(a). That is, the polynomial p(a) is a product of two other polynomials g(a) and h(a).
There are various methods of factorising a polynomial. They are,
1. Factorisation by dividing the expression by the HCF of the terms of the given expression.
2. Factorisation by grouping the terms of the expression.
3. Factorisation using identities.
Factorization using Identities
Recall the following identities for finding the products:
Factorization of trinomials
The general form of the trinomial is (x2 + cx + d) where c and d have different numerical values: c = a + b, and d = ab.
In the given trinomial expression if all terms are positive, then both the factors are positive.
If the middle term is negative, and last term is positive, signs of both the factors will be negative.
If the middle term is positive, and the last term is negative, the sign of one of the factors is positive and the other is negative.
Remainder Theorem
If f(x) is a polynomial in x and is divided by x-a; the remainder is the value of f(x) at x = a i.e., Remainder = f(a).
Factor Theorem
If p(x), a polynomial in x is divided by x-a and the remainder = f (a) is zero, then (x-a) is a factor of p(x).
Summary
An algebraic expression of the form a0+a1x+a2x2+….+anxn where
a0, a1, a2,….an are real numbers, n is a positive integer is called a polynomial in x.
