Class-9 : Maths - Chapter : 2. Polynomials. (Ex-2.1, Ex-2.3, Ex-2.3, Ex-2.4, Ex-2.5)

Points to Remember 

• Constants : A symbol having a fixed numerical value is called a constant. E.g : 9, -6, 4/7, √3, π

• Variables : A symbol which may be assigned different numerical values is known as a Variable. E.g : P = 4x , Here P & x are Variables & 4 is constant.

• 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. E.g : 5x² - 9x + 4 is an algebraic expression containing three algebraic terms, namely 5x², -9x & 4

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

• Classification of polynomials on the basis of degree :

Degree	Polynomial	Example
a). 1	Linear	(x+1) , (2x+3) etc.
b). 2	Quadratic	(ax² + bx + c) etc.
c). 3	Cubic	(x³ + 3x² + 7x + 4) etc.
d). 4	Biquadratic	(x⁴ - 1) etc.

• Classification of polynomials on the basis of number of terms :

No. of Terms	Polynomials	Example
a).  1	Monomial	-5, 3x, ⅛y,  etc.
b).  2	Binomial	(3x+6), (x-5p), etc.
c).  3	Trinomial	2x² + 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(α)=0, then we say that "α" 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 p(x) be a polynomial of degree 1 or more and let α be any real number. When p(x) is divided by (x-α) then the remainder is p(α).

• Factor theorem : Let p(x) be a polynomial of degree 1 or more and let α be any real number.

        (i)   If p(α) = 0 then, (x – α) is factor of f(x)

        (ii)  If (x – a) is factor of p(x) then p(α) = 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 = x² + 2xy + y²

       (ii) (x-y)² = x² - 2xy + y²

       (iii) x²-y² = (x+y) (x-y) 

       (iv) (x+a) (x+b) = x² + (a+b)x + ab

       (v)  (x+y+z)² = x² + y² + z² + 2xy+2yz+2zx

       (vi) (x+y)³ = x³ + y³ + 3xy(x+y)

       (vii) (x-y)³ = x³ - y³ - 3xy(x-y)

       (Viii) x³+y³ = (x+y) (x² - xy + y²)

       (ix) x³-y³ = (x-y) (x² + xy + y²)

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

           If, x+y+z=0 then, x³+y³+z³ = 3xyz

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Exercise - 2.1

Exercise -2.2

Exercise 2.3

Exercise - 2.4

Exercise - 2.5

To download Ch-2(Polynomials) solutions in PDF please click on the link below 

Exercise - 2.1 : Download PDF

Exercise - 2.2 : Download PDF

Exercise - 2.3 : Download PDF

Exercise - 2.4 : Download PDF

Exercise - 2.5 : Download PDF

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