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9 Mathematics notes Chapter 2 Polynomials

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CBSE Class 09 Mathematics
Revision Notes
CHAPTER – 2
POLYNOMIALS

Polynomials in one Variable
Zeroes of a Polynomial
Remainder Theorem
Factorisation of Polynomials
Algebraic Identities

Constants : A symbol having a fixed numerical value is called a constant.

Variables : A symbol which may be assigned different numerical values is known as variable.

Algebraic expressions : A combination of constants and variables connected by some or all of the operations +, -, *,/ is known as algebraic expression.

Terms : The several parts of an algebraic expression separated by ‘+’ or ‘-‘ operations are called the terms of the expression.

Polynomials : An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial.

(i) $5{x^2} - 4{x^2} - 6x - 3$ is a polynomial in variable x.

(ii) $5 + 8{x^{\frac{3}{2}}} + 4{x^{ - 2}}$ is an expression but not a polynomial.

Polynomials are denoted by p(x), q(x) and r(x) etc.

Coefficients : In the polynomial ${x^3} + 3{x^2} + 3x + 1,$ , coefficient of ${x^3},\;{x^2},\;x\;are\;1,\;3,\;3$ respectively and we also say that +1 is the constant term in it.

Degree of a polynomial in one variable: In case of a polynomial in one variable the highest power of the variable is called the degree of the polynomial.

A polynomial of degree n has n roots.

Classification of polynomials on the basis of degree.

degree	Polynomial	Example
(a) 1	Linear	$x + 1,\;2x + 3\,etc.$
(b) 2	Quadratic	$a{x^2} + bx + c\;\,etc.$
(c) 3	Cubic	${x^3} + 3{x^2} + 1\;\;\,etc.$
(d) 4	Biquadratic	${x^4} - 1$

Classification of polynomials on the basis of number of terms

No. of terms	Polynomial & Examples.
(i) 1	Monomial – $5,3x,\frac{1}{3}y\;etc.$
(ii) 2	Binomial – $(3 + 6x),\;(x - 5y)$ etc.
(iii) 3	Trinomial- $2{x^2} + 4x + 2\;etc.$

Constant polynomial : A polynomial containing one term only, consisting a constant term is called a constant polynomial.The degree of non-zero constant polynomial is zero.

Zero polynomial : A polynomial consisting of one term, namely zero only is called a zero polynomial.

The degree of zero polynomial is not defined.

Zeroes of a polynomial : Let $p(x)$ be a polynomial. If $p(a )$ =0, then we say that “a” is a zero of the polynomial of p(x).

Remark : Finding the zeroes of polynomial p(x) means solving the equation p(x)=0.

Remainder theorem : Let $f(x)$ be a polynomial of degree $n \geqslant 1$ and let a be any real number. When f(x) is divided by $(x - a)$ then the remainder is f ( a)

Factor theorem : Let f(x) be a polynomial of degree n > 1 and let a be any real number.

If f(a) = 0 then, (x – a) is factor of f(x)

If f(x – a) is factor of f(x) then f(a) = 0

Factor : A polynomial $p(x)$ is called factor of $q(x)$ divides $q(x)$ exactly.

Factorization : To express a given polynomial as the product of polynomials each of

degree less than that of the given polynomial such that no such a factor has a factor of

lower degree, is called factorization.

Some algebraic identities useful in factorization:

(i) ${(x+y)^2}$ = ${(x)^2}$ +2xy+ ${(y)^2}$

(ii) ${(x-y)^2}$ = ${(x)^2}$ -2xy+ ${(y)^2}$

(iii) ${x^2}$ – ${y^2}$ =(x-y)(x+y)

(iv) (x+a)(x+b)= ${(x)^2}$ +(a+b)x+ab

(v) ${(x+y+z)^2}$ = ${x^2}+$ ${y^2}+$ ${z^2}$ +2xy+2yz+2zx

(vi) ${(x+y)^3}$ = ${x^3}$ + ${y^3}$ +3xy(x+y)

(vii) ${(x-y)^3}$ = ${x^3}$ – ${y^3}$ -3xy(x-y)

(viii) ${x^3}$ + ${y^3}$ + ${z^3}$ – 3xyz =(x+y+z) ( ${x^2}$ + ${y^2}$ + ${z^2}$ – xy – yz – zx)

${x^3}$ + ${y^3}$ + ${z^3}$ =3xyz if x+y+z=0

(ix) ${a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$

(x) ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$

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