NCERT Textbook Chapter 2 - Polynomials, Class 10 (X) Mathematics Notes - Class 10

 Page 1


20 MATHEMA TICS
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
uu u ++ - is a polynomial
in the variable u of degree 6. Expressions like 
1
1 x -
, 2 x + , 
2
1
23 xx ++
 etc., are
not polynomials.
A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3,
35, x + 2 y + , 
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
uv vz -+ - + 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
POL YNOMIALS
Page 2


20 MATHEMA TICS
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
uu u ++ - is a polynomial
in the variable u of degree 6. Expressions like 
1
1 x -
, 2 x + , 
2
1
23 xx ++
 etc., are
not polynomials.
A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3,
35, x + 2 y + , 
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
uv vz -+ - + 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
POL YNOMIALS
POLYNOMIALS 21
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.
Page 3


20 MATHEMA TICS
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
uu u ++ - is a polynomial
in the variable u of degree 6. Expressions like 
1
1 x -
, 2 x + , 
2
1
23 xx ++
 etc., are
not polynomials.
A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3,
35, x + 2 y + , 
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
uv vz -+ - + 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
POL YNOMIALS
POLYNOMIALS 21
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.
22 MATHEMA TICS
Consider first a linear polynomial ax + b, a ? 0. Y ou 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
Page 4


20 MATHEMA TICS
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
uu u ++ - is a polynomial
in the variable u of degree 6. Expressions like 
1
1 x -
, 2 x + , 
2
1
23 xx ++
 etc., are
not polynomials.
A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3,
35, x + 2 y + , 
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
uv vz -+ - + 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
POL YNOMIALS
POLYNOMIALS 21
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.
22 MATHEMA TICS
Consider first a linear polynomial ax + b, a ? 0. Y ou 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
POLYNOMIALS 23
Table 2.1
x – 2 –1 012 3 4 5
y = x
2
 – 3x – 4 6 0 – 4– 6– 6 – 4 0 6
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
Page 5


20 MATHEMA TICS
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
uu u ++ - is a polynomial
in the variable u of degree 6. Expressions like 
1
1 x -
, 2 x + , 
2
1
23 xx ++
 etc., are
not polynomials.
A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3,
35, x + 2 y + , 
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
uv vz -+ - + 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
POL YNOMIALS
POLYNOMIALS 21
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.
22 MATHEMA TICS
Consider first a linear polynomial ax + b, a ? 0. Y ou 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
POLYNOMIALS 23
Table 2.1
x – 2 –1 012 3 4 5
y = x
2
 – 3x – 4 6 0 – 4– 6– 6 – 4 0 6
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
24 MATHEMA TICS
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.