The document discusses polynomials. Some key points: - A polynomial is an algebraic expression with variables that have only non-negative integral powers. - The highest power of a variable in a polynomial is called its degree. - Polynomials can be classified by the number of terms they contain (monomial, binomial, trinomial) or by their degree (linear, quadratic, cubic). - The remainder theorem states that when a polynomial is divided by (x-a), the remainder is equal to the value of the polynomial at x=a.

1. CLASS – IX
POLYNOMIALS
Name: Sandeep Jaiswal
Reg. no.: 11813483
Integrated. B.sc B.Ed.
Lovely Professional University

2. Algebraic Expression : The combination of constants and
variables are called algebraic expressions.
E.g.- (a) 2x3–4x2+6x–3 is a polynomial in one variable x.
(b) 8p7+4p2+11p3-9p is a polynomial in one variable p.
(c) 4+7x4/5+9x5 is an expression but not a polynomial
since it contains a term x4/5, where 4
/
5 is not
a non-negative integer.

3. Polynomials : An algebraic expression in which the
variable involved have only non –negative integral
powers is called a polynomial.
Eg. : 3x + 4y
Constants : A symbol having a fixed numerical value is
called a constant.
Eg. : In polynomial 3x + 4y ,3 and 4 are the constants.
Variables : A symbol which may be assigned different
numerical values is called as a variables.
Eg. : In polynomial 3x + 4y , x and y are the variables.

4. Degree : The highest power of a variable in the
polynomial is called degree of that polynomial.
Eg. : 5x2 + 3 , here the degree is 2.
Constant polynomial : A polynomial containing one
term only , consisting of a constant is called a
constant polynomial.
The degree of a nonzero constant polynomial is zero.
Eg. : 3 , -5 , 7/8 , etc. , are all constant polynomials.
Zero polynomial : A polynomial consisting one term
only , namely zero only , is called a zero polynomial.
The degree of a zero polynomial is not defined.

5. Monomial : Algebric expression that consists only one
term is called monomial.
Binomial : Algebric expression that consists two terms
is called binomial.
Trinomial : Algebric expression that consists three
terms is called trinomial.
Polynomial : Algebric expression that consists many
terms is called polynomial.

6. Types of polynomial on the basis of degree are :
Linear polynomial: A polynomial of degree 1 is called
a linear polynomial.
Quadratic polynomial: A polynomial of degree 2 is
called a quadratic polynomial.
Cubic polynomial : A polynomial of degree 3 is called
a cubic polynomial.
Biquadratic polynomial : A polynomial of degree 4 is
called a biquadratic polynomial.

7. POLYNOMI
ALS
Degree
5 0
Classify by
degree
Constant
Classify by no.
of terms.
Monomial
2x - 4 1 Linear Binomial
3x2 + x 2 Quadratic Binomial
x3 - 4x2 + 1 3 Cubic Trinomial

8. Phase 1 Phase 2
To rewrite a
polynomial in
standard form,
rearrange the
terms of the
polynomial
starting with the
largest degree
term and ending
with the lowest
degree term.
The leading coefficient,
the coefficient of the
first term in a
polynomial written in
standard form, should
be positive.
HOW TO CONVERT A POLYNOMIAL INTO
STANDARD FORM?

9. Let f(x) be a polynomial of degree n > 1 and let a be any real number.
When f(x) is divided by (x-a) , then the remainder is f(a).
PROOF Suppose when f(x) is divided by (x-a), the quotient is g(x) and the remainder is r(x).
Then, degree r(x) < degree (x-a)
degree r(x) < 1 [ therefore, degree (x-a)=1] degree
r(x) = 0
r(x) is constant, equal to r (say)
Thus, when f(x) is divided by (x-a), then the quotient is g9x) and the remainder is r.
Therefore, f(x) = (x-a)*g(x) + r (i)
Putting x=a in (i), we get r = f(a)
Thus, when f(x) is divided by (x-a), then the remainder is f(a).

10. Let f(x) be a polynomial of degree n > 1 and let a be
any real number.
(i) If f(a) = 0 then (x-a) is a factor of f(x).
PROOF let f(a) = 0
On dividing f(x) by 9x-a), let g(x) be the quotient. Also, by
remainder theorem, when f(x) is divided by (x-a), then
the remainder is f(a).
therefore f(x) = (x-a)*g(x) + f(a)
f(x) = (x-a)*g(x) [therefore f(a)=0(given]
(x-a) is a factor of f(x).

11. SOME COMMON IDENTITIES USED TO FACTORIZE
POLYNOMIALS
(x+a)(x+b)=x2+(a+b)x+ab
(a+b)2=a2+b2+2ab (a-b)2=a2+b2-2ab a2-b2=(a+b)(a-b)

12. ADVANCED IDENTITIES USED TO FACTORIZE
POLYNOMIALS
(x+y+z)2=x2+y2+z2
+2xy+2yz+2zx
(x-y)3=x3-y3-
3xy(x-y)
(x+y)3=x3+y3+
3xy(x+y)
x3+y3=(x+y) *
(x2+y2-xy) x3-y3=(x+y) *
(x2+y2+xy)