2. In class VIII you have learnt about algebraic
expressions and polynomials in one variable. In
this chapter, we strengthen our knowledge of
operations like Addition, Subtraction, Multiplication
and Division.
In addition to that, we also study how to factorise
some algebraic expressions with the help of some
Identities.
We shall also study about the Remainder Theorem
and Factor Theorem and their use in the
factorisation of Polynomials.

3. Monomial: A polynomial which is having one
term is called a monomial. Ex: 2xyz,6bc2
,…
Binomial: A polynomial which is having two
terms is called a binomial. Ex: 2x+5y, 4xyz-
3abc,…
Trinomial: A polynomial which is having
three terms is called a trinomial. Ex:
12ab+13ca- 10abc,…

4. A polynomial of degree one is called Linear
polynomial.
A polynomial of degree two is called Quadratic
polynomial.
A polynomial of degree three is called Cubic
polynomial.

5. Constant: A constant is a number or
an alphabet which remains constantly
with the variable.
Variable: A variable is an alphabet
which changes its value when the
equation changes.

6. We use distributive laws for multiplication
of polynomials.
We can solve the multiplication of
polynomials in two methods. We can solve
them by directly or by using column method.
Note: The degree of the product = Sum of
the degrees of the multiplicand and the
multiplier.

7. We have observed the multiplication of
polynomials. In case of division of
polynomials, the degree of the quotient is
equal to the degree of the dividend minus (-),
the degree of the divisor.
The remainder may be zero or its degree is at
least one less than that of the divisor.

9. Let p(x) be any polynomial with
degree greater than or equal to one
and let a be any real number.
If p(x) is divided by the linear
polynomial x – a, then the remainder
is p(a).

10. Let p(x) be any polynomial with degree greater than or equal
to 1. Suppose that when p(x) is divided by x – a, the quotient
is q(x) and the remainder is r(x), i.e.,
p(x) = (x – a) q(x) + r(x)
Since, the degree of x – a is 1 and the degree of r(x) is less
than the degree of x – a, the degree of r(x) = 0. This means
that r(x) is a constant, say r.
Therefore, p(x) = (x – a) q(x) + r(x)
In particular, if x=a, this equation gives us
p(a) = (a-a) q(a) + r(a) = r,
which proves the theorem.

11. If p(x) is a polynomial of degree
n>1 and a is any real number,
then (i) x – a is a factor of p(x),
if p(a) is 0, and
(ii) p(a) is 0, if x – a is a factor
of p(x).

12. By the remainder theorem, p(x) = (x – a)
q(x) + p(a).
(i) If p(a) = 0, then p(x)=(x – a) q(x),
which shows that x – a is a factor of
p(x).
(ii) Since x – a is a factor of p(x), p(x) =
(x – a) g(x) for same polynomial g(x).
In this case, p(a) = (a – a) g(a) = 0.

13. (x+y)2
= x2
+ 2xy + y2
(x+y)2 = x2
+ 2xy +
y2

14. (x+a) (x+b) = x2
+
(a+b) x + ab
x2
– y2
= (x + y) (x – y)

15. (x+y+z) = x2
+y2
+ z2
+ 2xy + 2yz +2zx
(x+y)3
= x3
+ y3
+ 3xy
(x+y)

16. (x-y)3
= x3
–y3
– 3xy(x-
y)
X3
+ y3
+ z3
-3xyz =
(x+y+z) (x2
+ y2
+ z2
–
xy–yz-zx

17. (a+b) (a2
-ab+b2
) = a3
+b3
(a-b) (a2
+ab-b2
) = a3
-b3

18. (a+b)3
= a3+3a2
b+3ab2
+b3
(a-b)3 = a3
-3a2
b+3ab2
-b3

19. (a+b+c)2
=
a2
+b2
+c2
+2ab+2bc+2ac
(a+b+c) (a2
+b2
+c2
-ab-
bc-ca) = a3
+b3
+c3
–
3abc

20. (ax+b) (cx+b) = ax (cx+d)
+b(cx+d)
(x+a) (x+b) (x+c) = (x+a)
[x2
+x(b+c)+bc