5. 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.

6. 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.

7. 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,…

8. 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.

9. 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.

10. 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.

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

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

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

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

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

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

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

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

20. 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).

21. 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.b

22. 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).

23. 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.