10 MATHEMATICS
2
2.1 Introduction
In Class IX, you have studied polynomials in one variable and their degrees. Recall
that if p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of
the polynomial p(x). For example, 4x + 2 is a polynomial in the variable x of
degree 1, 2y
2
– 3y + 4 is a polynomial in the variable y of degree 2, 5x
3
– 4x
2
+ x –
2
is a polynomial in the variable x of degree 3 and 7u
6
–
42
3
48
2
uuu
is a polynomial
in the variable u of degree 6. Expressions like
1
1x
,
2x
,
2
1
23xx
etc., are
not polynomials.
A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3,
35,x
2y
,
2
11
x
,
3z + 4,
2
1
3
u
, etc., are all linear polynomials. Polynomials
such as 2x + 5 – x
2
, x
3
+ 1, etc., are not linear polynomials.
A polynomial of degree 2 is called a quadratic polynomial. The name ‘quadratic’
has been derived from the word ‘quadrate’, which means ‘square’.
2
2
,
23
5
xx
y
2
– 2,
2
23,
x
x
22 2
21
25,5 ,4
337
u
uvvz
are some examples of
quadratic polynomials (whose coefficients are real numbers). More generally, any
quadratic polynomial in x is of the form ax
2
+ bx + c, where a, b, c are real numbers
and a 0. A polynomial of degree 3 is called a cubic polynomial. Some examples of
POLYNOMIALS
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POLYNOMIALS 11
a cubic polynomial are 2 – x
3
, x
3
,
3
2,
x
3 – x
2
+ x
3
, 3x
3
– 2x
2
+ x – 1. In fact, the most
general form of a cubic polynomial is
ax
3
+ bx
2
+ cx + d,
where, a, b, c, d are real numbers and a 0.
Now consider the polynomial p(x) = x
2
– 3x – 4. Then, putting x = 2 in the
polynomial, we get p(2) = 2
2
– 3 × 2 – 4 = – 6. The value ‘– 6’, obtained by replacing
x by 2 in x
2
– 3x – 4, is the value of x
2
– 3x – 4 at x = 2. Similarly, p(0) is the value of
p(x) at x = 0, which is – 4.
If p(x) is a polynomial in x, and if k is any real number, then the value obtained by
replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k).
What is the value of p(x) = x
2
–3x – 4 at x = –1? We have :
p(–1) = (–1)
2
–{3 × (–1)} – 4 = 0
Also, note that p(4) = 4
2
– (3 4) – 4 = 0.
As p(–1) = 0 and p(4) = 0, –1 and 4 are called the zeroes of the quadratic
polynomial x
2
– 3x – 4. More generally, a real number k is said to be a zero of a
polynomial p(x), if p(k) = 0.
You have already studied in Class IX, how to find the zeroes of a linear
polynomial. For example, if k is a zero of p(x) = 2x + 3, then p(k) = 0 gives us
2k + 3 = 0, i.e., k =
3
2
In general, if k is a zero of p(x) = ax + b, then p(k) = ak + b = 0, i.e.,
b
k
a
So, the zero of the linear polynomial ax + b is
(Constant term)
Coefficient of
b
ax
.
Thus, the zero of a linear polynomial is related to its coefficients. Does this
happen in the case of other polynomials too? For example, are the zeroes of a quadratic
polynomial also related to its coefficients?
In this chapter, we will try to answer these questions. We will also study the
division algorithm for polynomials.
2.2 Geometrical Meaning of the Zeroes of a Polynomial
You know that a real number k is a zero of the polynomial p(x) if p(k) = 0. But why
are the zeroes of a polynomial so important? To answer this, first we will see the
geometrical representations of linear and quadratic polynomials and the geometrical
meaning of their zeroes.
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12 MATHEMATICS
Consider first a linear polynomial ax + b, a 0. You have studied in Class IX that the
graph of y = ax + b is a straight line. For example, the graph of y = 2x + 3 is a straight
line passing through the points (– 2, –1) and (2, 7).
x –2 2
y = 2x + 3 –1 7
From Fig. 2.1, you can see
that the graph of y = 2x + 3
intersects the x-axis mid-way
between x = –1 and x = – 2,
that is, at the point
3
,
0
2
.
You also know that the zero of
2x + 3 is
3
2
. Thus, the zero of
the polynomial 2x + 3 is the
x-coordinate of the point where the
graph of y = 2x + 3 intersects the
x-axis.
In general, for a linear polynomial ax + b, a 0, the graph of y = ax + b is a
straight line which intersects the x-axis at exactly one point, namely,
,
0
b
a
.
Therefore, the linear polynomial ax + b, a 0, has exactly one zero, namely, the
x-coordinate of the point where the graph of y = ax + b intersects the x-axis.
Now, let us look for the geometrical meaning of a zero of a quadratic polynomial.
Consider the quadratic polynomial x
2
– 3x – 4. Let us see what the graph* of
y = x
2
– 3x – 4 looks like. Let us list a few values of y = x
2
– 3x – 4 corresponding to
a few values for x as given in Table 2.1.
* Plotting of graphs of quadratic or cubic polynomials is not meant to be done by the students,
nor is to be evaluated.
Fig. 2.1
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POLYNOMIALS 13
Table 2.1
x – 2–10123
45
y = x
2
– 3x – 46 0– 4– 6– 6– 4 06
If we locate the points listed
above on a graph paper and draw
the graph, it will actually look like
the one given in Fig. 2.2.
In fact, for any quadratic
polynomial ax
2
+ bx + c, a 0, the
graph of the corresponding
equation y = ax
2
+ bx + c has one
of the two shapes either open
upwards like
or open
downwards like
depending on
whether a > 0 or a < 0. (These
curves are called parabolas.)
You can see from Table 2.1
that –1 and 4 are zeroes of the
quadratic polynomial. Also
note from Fig. 2.2 that –1 and 4
are the x-coordinates of the points
where the graph of y = x
2
– 3x – 4
intersects the x-axis. Thus, the
zeroes of the quadratic polynomial
x
2
– 3x – 4 are x-coordinates of
the points where the graph of
y = x
2
– 3x – 4 intersects the
x-axis.
This fact is true for any quadratic polynomial, i.e., the zeroes of a quadratic
polynomial ax
2
+ bx + c, a 0, are precisely the x-coordinates of the points where the
parabola representing y = ax
2
+ bx + c intersects the x-axis.
From our observation earlier about the shape of the graph of y = ax
2
+ bx + c, the
following three cases can happen:
Fig. 2.2
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14 MATHEMATICS
Case (i) : Here, the graph cuts x-axis at two distinct points A and A.
The x-coordinates of A and A are the two zeroes of the quadratic polynomial
ax
2
+ bx + c in this case (see Fig. 2.3).
Fig. 2.3
Case (ii) : Here, the graph cuts the x-axis at exactly one point, i.e., at two coincident
points. So, the two points A and A of Case (i) coincide here to become one point A
(see Fig. 2.4).
Fig. 2.4
The x-
coordinate of A is the only zero for the quadratic polynomial ax
2
+ bx + c
in this case.
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POLYNOMIALS 15
Case (iii) : Here, the graph is either completely above the x-axis or completely below
the x-axis. So, it does not cut the x-axis at any point (see Fig. 2.5).
Fig. 2.5
So, the quadratic polynomial ax
2
+ bx + c has no zero in this case.
So, you can see geometrically that a quadratic polynomial can have either two
distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This also means that a
polynomial of degree 2 has atmost two zeroes.
Now, what do you expect the geometrical meaning of the zeroes of a cubic
polynomial to be? Let us find out. Consider the cubic polynomial x
3
– 4x. To see what
the graph of y = x
3
– 4x looks like, let us list a few values of y corresponding to a few
values for x as shown in Table 2.2.
Table 2.2
x –2 –1 0 1 2
y = x
3
– 4x 030 –3 0
Locating the points of the table on a graph paper and drawing the graph, we see
that the graph of y = x
3
– 4x actually looks like the one given in Fig. 2.6.
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16 MATHEMATICS
We see from the table above
that – 2, 0 and 2 are zeroes of the
cubic polynomial x
3
– 4x. Observe
that –2, 0 and 2 are, in fact, the
x-coordinates of the only points
where the graph of y = x
3
– 4x
intersects the x-axis. Since the curve
meets the x-axis in only these 3
points, their x-coordinates are the
only zeroes of the polynomial.
Let us take a few more
examples. Consider the cubic
polynomials x
3
and x
3
– x
2
. We draw
the graphs of y = x
3
and y = x
3
– x
2
in Fig. 2.7 and Fig. 2.8 respectively.
Fig. 2.7 Fig. 2.8
Fig. 2.6
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POLYNOMIALS 17
Note that 0 is the only zero of the polynomial x
3
. Also, from Fig. 2.7, you can see
that 0 is the x-coordinate of the only point where the graph of y = x
3
intersects the
x-axis. Similarly, since x
3
– x
2
= x
2
(x – 1), 0 and 1 are the only zeroes of the polynomial
x
3
– x
2
. Also, from Fig. 2.8, these values are the x-coordinates of the only points
where the graph of y = x
3
– x
2
intersects the x-axis.
From the examples above, we see that there are at most 3 zeroes for any cubic
polynomial. In other words, any polynomial of degree 3 can have at most three zeroes.
Remark : In general, given a polynomial p(x) of degree n, the graph of y = p(x)
intersects the x-axis at atmost n points. Therefore, a polynomial p(x) of degree n has
at most n zeroes.
Example 1 : Look at the graphs in Fig. 2.9 given below. Each is the graph of y = p(x),
where p(x) is a polynomial. For each of the graphs, find the number of zeroes of p(x).
Fig. 2.9
Solution :
(i) The number of zeroes is 1 as the graph intersects the x-axis at one point only
.
(ii) The number of zeroes is 2 as the graph intersects the x-axis at two points.
(iii) The number of zeroes is 3. (Why?)
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18 MATHEMATICS
(iv) The number of zeroes is 1. (Why?)
(v) The number of zeroes is 1. (Why?)
(vi) The number of zeroes is 4. (Why?)
EXERCISE 2.1
1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the
number of zeroes of p(x), in each case.
Fig. 2.10
2.3 Relationship between Zeroes and Coefficients of a Polynomial
You have already seen that zero of a linear polynomial ax + b is
b
a
. We will now try
to answer the question raised in Section 2.1 regarding the relationship between zeroes
and coefficients of a quadratic polynomial. For this, let us take a quadratic polynomial,
say p(x) = 2x
2
– 8x + 6. In Class IX, you have learnt how to factorise quadratic
polynomials by splitting the middle term. So, here we need to split the middle term
‘– 8x’ as a sum of two terms, whose product is 6 × 2x
2
= 12x
2
. So, we write
2x
2
– 8x + 6 = 2x
2
– 6x – 2x + 6 = 2x(x – 3) – 2(x – 3)
=(2x – 2)(x – 3) = 2(x – 1)(x – 3)
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POLYNOMIALS 19
So, the value of p(x) = 2x
2
– 8x + 6 is zero when x – 1 = 0 or x – 3 = 0, i.e., when
x = 1 or x = 3. So, the zeroes of 2x
2
– 8x + 6 are 1 and 3. Observe that :
Sum of its zeroes
=
2
( 8) (Coefficient of )
134
2
Coefficient of
x
x
Product of its zeroes
=
2
6 Constant term
133
2
Coefficient of
x
Let us take one more quadratic polynomial, say, p(x) = 3x
2
+ 5x – 2. By the
method of splitting the middle term,
3x
2
+ 5x
–
2 = 3x
2
+ 6x – x – 2 = 3x(x + 2) –1(x + 2)
=(3x – 1)(x + 2)
Hence, the value of 3x
2
+ 5x – 2 is zero when either 3x – 1 = 0 or x + 2 = 0, i.e.,
when x =
1
3
or x = –2. So, the zeroes of 3x
2
+ 5x – 2 are
1
3
and – 2. Observe that :
Sum of its zeroes =
2
1 5 (Coefficient of )
(2)
33
Coefficient of
x
x
Product of its zeroes =
2
1 2 Constant term
(2)
33
Coefficient of
x
In general, if * and * are the zeroes of the quadratic polynomial p(x) = ax
2
+ bx + c,
a 0, then you know that x – and x – are the factors of p(x). Therefore,
ax
2
+ bx + c = k(x – ) (x – ), where k is a constant
= k[x
2
– ( + )x + ]
= kx
2
– k( + )x + k
Comparing the coefficients of x
2
, x and constant terms on both the sides, we get
a = k, b =– k( + ) and c = k
This gives
+
=
–
b
a
,
=
c
a
* , are Greek letters pronounced as ‘alpha’ and ‘beta’ respectively. We will use later one
more letter ‘’ pronounced as ‘gamma’.
Rationalised 2023-24
20 MATHEMATICS
i.e., sum of zeroes = + =
2
(Coefficient of )
Coefficient of
bx
a
x
,
product of zeroes = =
2
Constant term
Coefficient of
c
a
x
.
Let us consider some examples.
Example 2 : Find the zeroes of the quadratic polynomial x
2
+ 7x + 10, and verify the
relationship between the zeroes and the coefficients.
Solution : We have
x
2
+ 7x + 10 = (x + 2)(x + 5)
So, the value of x
2
+ 7x + 10 is zero when x + 2 = 0 or x + 5 = 0, i.e., when x = – 2 or
x = –5. Therefore, the zeroes of x
2
+ 7x + 10 are – 2 and – 5. Now,
sum of zeroes =
2
(7) –(Coefficient of )
,
–2 (–5) –(7)
1
Coefficient of
x
x
product of zeroes =
2
10 Constant term
(2) (5) 10
1
Coefficient of
x
Example 3 : Find the zeroes of the polynomial x
2
– 3 and verify the relationship
between the zeroes and the coefficients.
Solution : Recall the identity a
2
– b
2
= (a – b)(a + b). Using it, we can write:
x
2
– 3 =
33xx
So, the value of x
2
– 3 is zero when x =
3
or x =
–3
Therefore, the zeroes of x
2
– 3 are
3
and
3
Now,
sum of zeroes =
2
(Coefficient of )
,
330
Coefficient of
x
x
product of zeroes =
2
3 Constant term
33–3
1
Coefficient of
x
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POLYNOMIALS 21
Example 4 : Find a quadratic polynomial, the sum and product of whose zeroes are
–3 and 2, respectively.
Solution : Let the quadratic polynomial be ax
2
+ bx + c, and its zeroes be and .
We have
+ = – 3 =
b
a
,
and =2 =
c
a
.
If a = 1, then b = 3 and c = 2.
So, one quadratic polynomial which fits the given conditions is x
2
+ 3x + 2.
You can check that any other quadratic polynomial that fits these conditions will
be of the form k(x
2
+ 3x + 2), where k is real.
Let us now look at cubic polynomials. Do you think a similar relation holds
between the zeroes of a cubic polynomial and its coefficients?
Let us consider p(x) = 2x
3
– 5x
2
– 14x + 8.
You can check that p(x) = 0 for x = 4, – 2,
1
2
Since p(x) can have atmost three
zeroes, these are the zeores of 2x
3
– 5x
2
– 14x + 8. Now,
sum of the zeroes =
2
3
1 5 ( 5) (Coefficient of )
4(2)
22 2
Coefficient of
x
x
,
product of the zeroes =
3
1 8 –Constant term
4(2) 4
22
Coefficient of
x
.
However, there is one more relationship here. Consider the sum of the products
of the zeroes taken two at a time. We have
11
4(2) (2) 4
22
=
14
–8 1 2 7
2
=
3
Coefficient of
Coefficient of
x
x
.
In general, it can be proved that if , , are the zeroes of the cubic polynomial
ax
3
+ bx
2
+ cx + d, then
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22 MATHEMATICS
+
+
=
–
b
a
,
+
+
=
c
a
,
=
–
d
a
.
Let us consider an example.
Example 5* : Verify that 3, –1,
1
3
are the zeroes of the cubic polynomial
p(x) = 3x
3
– 5x
2
– 11x – 3, and then verify the relationship between the zeroes and the
coefficients.
Solution : Comparing the given polynomial with ax
3
+ bx
2
+ cx + d, we get
a = 3, b = – 5, c = –11, d = – 3. Further
p(3) = 3 × 3
3
–
(5 × 3
2
) – (11 × 3) – 3 = 81 – 45 – 33 – 3 = 0,
p(–1) = 3 × (–1)
3
– 5 × (–1)
2
– 11 × (–1) – 3 = –3 – 5 + 11 – 3 = 0,
32
11 1 1
35113
33 3 3
p
,
=
1511 22
–3–0
99 3 33
Therefore, 3, –1 and
1
3
are the zeroes of 3x
3
– 5x
2
– 11x – 3.
So, we take = 3, = –1 and =
1
3
Now,
115(5)
3(1) 2
3333
b
a
,
11 1 11
3(1)(1) 3 3 1
33 33
c
a
,
1(3)
3(1) 1
33
d
a
.
* Not from the examination point of view.
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POLYNOMIALS 23
EXERCISE 2.2
1. Find the zeroes of the following quadratic polynomials and verify the relationship between
the zeroes and the coefficients.
(i) x
2
– 2x – 8 (ii) 4s
2
– 4s + 1 (iii) 6x
2
– 3 – 7x
(iv) 4u
2
+ 8u (v) t
2
– 15 (vi) 3x
2
– x – 4
2. Find a quadratic polynomial each with the given numbers as the sum and product of its
zeroes respectively.
(i)
1
,
1
4
(ii)
1
2,
3
(iii)
0, 5
(iv) 1, 1 (v)
11
,
44
(vi) 4, 1
2.4 Summary
In this chapter, you have studied the following points:
1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials
respectively.
2. A quadratic polynomial in x with real coefficients is of the form ax
2
+ bx + c, where a, b, c
are real numbers with a 0.
3. The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the
graph of y = p(x) intersects the x-axis.
4. A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have
at most 3 zeroes.
5. If and are the zeroes of the quadratic polynomial ax
2
+ bx + c, then
b
a
,
c
a
.
6. If , , are the zeroes of the cubic polynomial ax
3
+ bx
2
+ cx + d, then
b
a
,
c
a
,
and
d
a
.
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