This document contains a summary of key concepts in polynomials including: 1) Definitions of terms like variable, term, coefficient, degree, constant and zero polynomials. 2) The Remainder Theorem and how it relates the remainder of polynomial division to the factor theorem. 3) The Factor Theorem and how it can be used to determine if a polynomial is a factor of another. 4) Examples of factoring polynomials and using the Factor Theorem. 5) A list of 15 common algebraic identities involving polynomials.

1. kendriya
vidyalaya

2. MATHS
PROJECT
WORK

4. 1.INTRODUCTION
2.POLYNOMIALS IN ONE VARIABLE
3.ZERO POLYNOMIAL
4.REMAINDER THEOREM
5. FACTOR THEOREM
6. FACTORISATION OF POLYNOMIALS
7.ALGEBRIC IDENTITIES

5. INTRODUCTION
We know about algebric
expressions, their additions,
subtraction, multiplication and
division in earlier classes. We
have studied how to factorise
some algebraic expressions.

6. Some identities are-:
(a+b)²=a²+b²+2ab
(a-b)²=a²+b²-2ab
a²-b²=(a+b)(a-b)

7. 2.POLYNOMIALS IN ONE VARIABLE
VARIABLE:- A variable is denoted
by a symbol that can take any real
value.We use letters like x, y, z to
denote variables.

8. 2x,3y,-x,-1/2x are all algebraic expressions.
All these expressions are of the form
(a constant).x. So we can say
it as (a constant)X(a variable) and we
and we don’t know what the constant
is. In such cases, we write the constant
as a, b, c etc. So the expression will be
ax.

9. TERM:- In the polynomial x² +2x ,
x² & 2x are called the terms.
COEFFICIENT:-Each term of a
polynomial has a coefficient. So, in
-x³+4x²+7x-2,the coefficient of
x³ is -1, the coefficient of x² is 4,
the coefficient of x is 7 and -2 is
the coefficient of x0 .

10. DEGREE:-The highest power
of variable in the polynomial
is known as degree of the
polynomial.
For ex:-5x2 +3 ,here the degree
is 2.

11. CONSTANT POLYNOMIAL:-A polynomial
containing one term only, consisting of
a constant is called a constant
polynomial.
The degree of a non zero constant
polynomial is zero.
Eg:-3, -5, 7/8,etc., are all
constant polynomials.

12. 3.ZERO POLYNOMIAL :-A
polynomial consisting one term
only, namelyzero only, is called
a zero polynomial.
The degree of a zero
polynomial is not defined.

15. 4.REMAINDER THEOREM :-We know that,
when a natural number n is divided by a natural number
m less than or equal to n, the remainder is either 0 or
a natural number r<m.
Example:23 when divided by 5 gives the quotient 4 and the
remainder 3. Here,we can express 23 as 23=(5x4)+3
i.e.,Dividend=(Divisor X Quotient)+Remainder
Now, we extend the above phenomenon
for the division of a polynomial p(x). Then,we
can find polynomial q(x) and r(x) such that:-
p(x)=g(x) X q(x)+ r(x),where r(x)=0 or degree or
r(x),degree of g(x).Division of a polynomial
by a linear polynomial.

16. 5. FACTOR THEOREM:-Let p(x) be
a polynomial of degree n≥1 and a
be any real constant then
If p(a) =0,then (x-a) is a factor of p(x).
P(x)=(x-a) X q(x) +p(a)

17. 6. FACTORISATION OF POLYNOMIALS
EXAMPLE
Question 2:
Use the Factor Theorem to determine whether g(x) is
a factor of p(x) in each of the following cases:
(i) p(x) = 2x3 + x2 − 2x − 1, g(x) = x + 1
(ii) p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2
(iii) p(x) = x3 − 4 x2 + x + 6, g(x) = x − 3
Answer :

18. (i) If g(x) = x + 1 is a factor of the given
polynomial p(x), then p(−1) must be zero.
p(x) = 2x3 + x2 − 2x − 1
p(−1) = 2(−1)3 + (−1)2 − 2(−1) − 1
= 2(−1) + 1 + 2 − 1 = 0
Hence, g(x) = x + 1 is a factor of the given polynomial.
(ii) If g(x) = x + 2 is a factor of the given
polynomial p(x), then p(−2) must
be 0.
p(x) = x3 +3x2 + 3x + 1
p(−2) = (−2)3 + 3(−2)2 + 3(−2) + 1
= − 8 + 12 − 6 + 1
= −1

19. As p(−2) ≠ 0,
Hence, g(x) = x + 2 is not a factor of the
given polynomial.
(iii) If g(x) = x − 3 is a factor of the given
polynomial p(x), then p(3) must
be 0.
p(x) = x3 − 4 x2 + x + 6
p(3) = (3)3 − 4(3)2 + 3 + 6
= 27 − 36 + 9 = 0
Hence, g(x) = x − 3 is a factor of the given
polynomial.

20. 7.ALGEBRIC IDENTITIES
1.(x + a)(x - b) = x2 + (a - b) x - ab
2. (x - a)(x + b) = x2 + (b - a) x - ab
3.(x - a)(x - b) = x2 - (a + b)x + ab
4.(a + b)3 = a3 + b3 +3ab (a + b)
5.(a - b)3 = a3 - b3 - 3ab (a - b)
z)2 + (z -x)2]

21. 6.(x + y + z)2 = x2 + y2 + z2 + 2xy +2yz +
2xz
7.(x + y - z)2 = x2 + y2 + z2 + 2xy - 2yz
– 2xz
8. (x - y + z)2 = x2 + y2 + z2 - 2xy - 2yz
+ 2xz
9. (x - y - z)2 = x2 + y2 + z2 - 2xy +
2yz - 2xz

22. 10.x3 + y3 + z3 - 3xyz = (x + y + z) (x2 + y2
+ z2 - xy - yz -xz)
11. x2 + y2 = 12 [(x+y)2 + (x-y)2]
12. (x+a) (x+b) (x+c) = x3+ (a + b + c )
x2 + (ab + bc +ca) x + abc
13. x3 + y3 = (x + y) ( x2 - xy + y2)
14. x3 - y3 = (x - y) ( x2 + xy + y2)
15. x2 + y2 + z2 -xy -yz -zx = 12
[(x-y)2 + (y -

23. SUBMITTED TO
P.K. SRIVASTAVA SIR
{THANX SIR FOR LIGHTING US WITH UR
KNOWLEDGE}

25. THE END
THANK YOU