MIND
Polynomial in One Variable
An algebraic expression which have only whole numbers as the exponent of one variable,
is called polynomial in one variable.
e.g. 3x 3 + 2x 2 – 7x + 5 etc.
Algebraic Expression Degree of a Polynomial Terms and Coefficient
A combination of constants and variables, connected Highest power of the variable in a The part of a polynomial separated from
by four fundamental arithmetical operations +, -, ´ polynomial, is known as degree of that each other by + or - sign is called a
and ¸ is called an algebraic expression. polynomial. term and each term of a polynomial has
e.g. 6x 2 - 5y 2 + 2xy a coefficient.
On the Basis of Degree of Variables
(i ) A polynomial of degree 0, is called a
Value of a Polynomial On the Basis of Number of Terms
constant polynomial.
(ii ) A polynomial of degree 1, is called a (i ) A polynomial containing one non-zero
The value obtained on putting a particular value of term,iscalleda monomial.
linear polynomial.
the variable in polynomial is called value of the (ii ) A polynomial containing two non-zero
polynomial at that value of variable. (iii ) A polynomial of degree 2, is called a
quadratic polynomial. terms,iscalleda binomial.
(iv ) A polynomial of degree 3, is called a (iii ) A polynomial containing three non-zero
cubic polynomial. terms,iscalleda trinomial.
(v) A polynomial of degree 4, is called a
Zero of a Polynomial biquadratic polynomial.
Zero of a polynomial p (x) is a number a, such Remainder Theorem
that p (a) = 0. It is also called root of polynomial
Let f (x) be any polynomial of degree
equation p (x) = 0. n degree Polynomial n,(n ³ 1) and a be any real number.
A polynomial in one variable x of degree n, If f (x) is divided by the linear
is an expression of the form p (x) = anx n polynomial (x – a), then the remainder
Factor Theorem + an – 1 x n –1 + ... + a2x 2 + a1x + a0, where an , is f (a).
Let f (x) be a polynomial of degree n, (n ³ 1) an – 1 ,..., a2, a1, a0 are constants and an ¹ 0.
and a be any real number. Then,
(i ) if f (a) = 0, then (x – a) is a factor of f (x).
(ii ) if (x – a) is a factor of f (x), then f (a) = 0.
Division Algorithm Algebraic Identities
(i ) (x + y) 2 = x 2 + y 2 + 2xy
If p (x) and g (x) are any two polynomials such
(ii ) (x – y) 2 = x 2 + y 2 – 2xy
that degree of p (x) ³ degree of g (x) and
(iii ) x 2 – y 2 = (x – y) (x + y)
g (x) ¹ 0, then we can find polynomials q (x) and r
(iv ) (x + a) (x + b) = x 2 + (a + b) x + ab
(x), such that p (x) = g (x).q (x) + r (x), i.e.
Factorisation of a Quadratic Polynomial (v) (x + y + z) 2= x 2+ y 2+ z 2 + 2xy + 2yz + 2zx
Dividend = (Divisor ´ Quotient) + Remainder,
A quadratic polynomial ax 2 + bx + c can be where r (x) = 0 or degree of r (x) < degree of g (x). (vi ) (x + y) 3 = x 3 + y 3 + 3xy (x + y)
factorise by two methods (vii ) (x – y) 3 = x 3 – y 3 – 3xy (x – y)
(i ) By splitting the middle term In this method, (viii ) x 3 – y 3 = (x – y) (x 2 + xy + y 2)
write b as the sum of two numbers (say p and (ix ) x 3 + y 3 = (x + y) (x 2 – xy + y 2)
q), whose product is ac, i.e.
(x ) x 3 + y 3 + z 3 – 3xyz
write b = p + q, such that pq = ac
Factorisation of a Cubic Polynomial = (x + y + z) (x 2+ y 2+ z 2– xy – yz –zx)
(ii ) By factor theorem Write the given polynomial
as ax 2 + bx + c = a [x 2+ (b /a) x + c /a] = a×p (xi ) x 2 + y 2 + z 2 – xy – yz – zx
For cubic polynomial, find atleast one factor first and 1
(x), where p (x) = x 2 + (b /a) x + c /a. Now, find then adjust the given polynomial such that it = — [(x – y) 2 + (y –z) 2 + (z – x) 2]
2
all possible factors of c/a and then by trial becomes the product of two terms (or expressions)
method, take that factor at which p (x) is zero. in which one of them is that factor and the other is a
quadratic polynomial.
