Value of a polynomial
If p(x) is a polynomial in x and if k is any real number, then the
value obtained by putting x = k in p(x), is called the value of the
polynomial p(x) at x = k.
The value of p(x) at x = k is denoted by p(k)
e.g. If p(x) = x² – 4x + 5
then,
Zeroes of a Polynomial
The real values of the variable for which the value of the
polynomial becomes zero are called zeroes of the polynomial.
e.g. p(x) = 2x3 – 3x2 – 2x + 3
, p(1) =
∴ 1 is a zero of p(x)
If x = 1
2(1)3
Replace x by 4
Replace x by 1
p(4)
=
(4)²
–
4(4)
+
5
=
16
–
16
+
5
=
5
–
3(1)2
–
2(1)
+ 3
=
2
–
3
–
2
+
3
=
0
Value of a polynomial become zero for x = 1
1 is called zero of a polynomial
Q. 1
EXERCISE 2.1
The graphs of y = p(x) are given below, for some polynomials p(x).
Find the number of zeroes of p(x), in each case.
X′
Y
X
Y′
O
Sol.
The number of zeroes is 0, because the graph does
not intersect the X - axis at any point.
Given line is intersecting X axis at how many points ?
Given line does not intersect the X axis at any point
The polynomial is in the form y = p(x),
For finding the number of zeroes for the given figures we have to just check the number of points where the graphs intersects the X axis
(i)
1 Mark
X′
Y
X
Y’
O
Sol.
The number of zeroes is 1, because the graph intersects
the X - axis at one point.
(ii)
Q. 1
EXERCISE 2.1
The graphs of y = p(x) are given below, for some polynomials p(x).
Find the number of zeroes of p(x), in each case.
1 Mark
The number of zeroes is 3, because the graph intersects
the X - axis at three points.
X′
Y
X
Y′
O
Sol.
(iii)
Q. 1
EXERCISE 2.1
The graphs of y = p(x) are given below, for some polynomials p(x).
Find the number of zeroes of p(x), in each case.
1 Mark
X′
Y
X
Y′
O
The number of zeroes is 2, because the graph intersects the X - axis at two points.
Sol.
(iv)
Q. 1
EXERCISE 2.1
The graphs of y = p(x) are given below, for some polynomials p(x).
Find the number of zeroes of p(x), in each case.
1 Mark
X′
Y
X
Y′
O
The number of zeroes is 4, because the graph intersects the X - axis at four points
Sol.
(v)
Q. 1
EXERCISE 2.1
The graphs of y = p(x) are given below, for some polynomials p(x).
Find the number of zeroes of p(x), in each case.
1 Mark
X′
Y
X
Y′
O
The number of zeroes is 3, because the graph intersects the X - axis at three points.
Sol.
(vi)
Q. 1
EXERCISE 2.1
The graphs of y = p(x) are given below, for some polynomials p(x).
Find the number of zeroes of p(x), in each case.
1 Mark