Polynomials - Revision Notes

 CBSE Class 09 Mathematics

Revision Notes

CHAPTER – 2

POLYNOMIALS

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1. Polynomials in one Variable
2. Zeroes of a Polynomial
3. Remainder Theorem
4. Factorisation of Polynomials
5. 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) 5x2−4x2−6x−3$5x^2-4x^2-6x-3$ is a polynomial in variable x.

(ii) 5+8x32+4x−2$5+8x^{\frac{3}{2}}+4x^{-2}$ is an expression but not a polynomial.

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

Coefficients : In the polynomial x3+3x2+3x+1,$x^3+3x^2+3x+1,$, coefficient of x3,x2,xare1,3,3$x^3,x^2,xare1,3,3$ 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+3etc.$x+1,2x+3etc.$
(b) 2	Quadratic	ax2+bx+cetc.$a{x}^2+bx+cetc.$
(c) 3	Cubic	x3+3x2+1etc.$x^3+3x^2+1etc.$
(d) 4	Biquadratic	x4−1$x^4-1$

Classification of polynomials on the basis of number of terms

No. of terms	Polynomial & Examples.
(i) 1	Monomial - 5,3x,13yetc.$5,3x,\frac{1}{3}yetc.$
(ii) 2	Binomial - (3+6x),(x−5y)$(3+6x),(x-5y)$ etc.
(iii) 3	Trinomial- 2x2+4x+2etc.$2x^2+4x+2etc.$

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)$p(x)$ be a polynomial. If p(a)$p(a)$=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)$f(x)$ be a polynomial of degree n⩾1$n⩾1$ and let a be any real number. When f(x) is divided by (x−a)$(x-a)$ 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)$p(x)$ is called factor of q(x)$q(x)$ divides q(x)$q(x)$ 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$(x+y{)}^2$= (x)2$(x{)}^2$+2xy+(y)2$(y{)}^2$

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

(iii) x2$x^2$-y2$y^2$=(x-y)(x+y)

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

(v)(x+y+z)2$(x+y+z{)}^2$=x2+$x^2+$y2+$y^2+$z2$z^2$+2xy+2yz+2zx

(vi) (x+y)3$(x+y{)}^3$=x3$x^3$+y3$y^3$+3xy(x+y)

(vii) (x−y)3$(x-y{)}^3$=x3$x^3$-y3$y^3$-3xy(x-y)

(viii) x3$x^3$ + y3$y^3$ + z3$z^3$ - 3xyz =(x+y+z) (x2$x^2$ + y2$y^2$ + z2$z^2$ - xy - yz - zx)

x3$x^3$+y3$y^3$+z3$z^3$=3xyz if x+y+z=0

(ix) a3+b3=(a+b)(a2−ab+b2)$a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$

(x) a3−b3=(a−b)(a2+ab+b2)