1) The document introduces concepts related to polynomials including constants, variables, terms, like terms, unlike terms, and different types of polynomials such as monomials, binomials, trinomials, and multinomials.
2) It discusses the degree of polynomials including linear, quadratic, and cubic polynomials. It also covers the value and zeros of polynomials.
3) The document explains important polynomial concepts such as the factor theorem, polynomial identities, and important points about zeros of polynomials.
3. Introduction
Constant :-component which never change its
value or magnitude is known as constant for
example all real no. Are always constant as they
never changes its values.
Variable :-component of any term or expression or
equation which varies situation is known as
variable.
Term :-term is an element which is combination of
4 signs , numbers , variable and power or a term
always has 4 things sign + or-
4. Types of terms
Like terms – two or more having
same type of variable and same power
on them are said to be like terms for
example 3x ,-7/2x,8/9x , are like terms.
Unlike terms –terms if they are not
like then they are known as unlike
terms for example 7a , 8b , 19/3c are
unlike terms.
5. What is polynomial ?
An algebraic expression in the form of : 2a2
+3b+5c+6x,…..+…….
Different types of polynomials:-
1.Monomial :-expression have single term.
2.Binomial :- expression have two terms.
3.Trinomial :-expression have three term.
4. Multinoamil :-expression have more than three
terms.
5.Zero polynomial:-number itself is known as
zero polynomial.
6. Degree of a polynomial
1.Linear polynomial :- A polynomial of the form ;ax +
b, a = 0 is known as linear polynomial its degree is
always zero it may be monomial or binomial . It may be
monomial or binomial for example each of polynomial
2x , -3x is a linear polynomial as well as monomial and
linear polynomial.
2. Quadratic polynomial :- an algebraic expression
of type ax2 +bx +c,a is not equal 0 is known as quadraic
polynomial, or we can say that polynomial of degree2 is
known as quadraic polynomial, quadratic polynomial
can be a monomial , binomial or trinomial.
7. 3. cubic polynomials – a polynomial
of the form of ax3+bx2+cx+d , a=0 is
known as cubic polynomial . A cubic
polynomial may be monomial ,
binomial , trinomial , multinomial .
8. VALUE AND ZEROES OF POLYNOMIAL
Value of a polynomial
The value of a polynomial p(x) at x = a is
p(a) . Obtained on replacing x by a .
Zeroes of a polynomial
In general we say that (a) is a zero of
polynomial p(x) at a such that p(a)=0 .
9. Factor theorem
Let p{x} be any polynomial of greater than or equal to 1 and “a”
be any real number , , then
i. {x-a} is a factor of p{x}, if p{a}=0;and
ii. P{a]=0 if {x-a}is a factor of p{x}.
iii. Proof :let p{x} be a polynomial of degree n >1 and “a” be a
real number.
iv. If p {a} =0 {given}
v. Let q{x] be the quotient when p{x} be divided by {x-a}.
vi. By reminder theorem , remainder =p{a}
vii. Polynomial= divisor* quoient +remainder
viii. p{x}={x-a} q{x}+p{a}=p{x}=[x-a]q[x]:p{a}=0 proved
10. IDENTITIES
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a +a2 – 2ab + b2
a2 – b2 = ( b)(a – b)
(x + a)(x + b) = x2 + (a + b)x + ab
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
(a + b)3 = a3 + b3 + 3ab (a + b)
(a – b)3 = a3 – b3 – 3ab (a – b)
a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)
a3 + b3 + c3 = 3abc , If a + b + c = 0
11. Important points to remember :
A constant polynomial does not has any
zero .
0 may be a zero of a polynomial .
Every linear polynomial has one and only
one zero .
A polynomial can have repeated zeroes .
Number of zeroes of a polynomial cannot
exceed its degree.
