Give examples to justify the following statement :
a) A zero of a polynomial need not be 0.
b) O may be a zero of a polynomial.
c) Every linear polynomial has one and only one zero.
d) A polynomial can have more than one zero.
Answer :
a ) A zero of a polynomial need not be 0.
that statement is right , because polynomial can have different zeros that can be zero or apart from that .
Example : we have a polynomial
f( x ) = 2x3 + 5x2 - 3x
we can write it As: x( 2x2 + 5x - 3 )
Equate equal to zero , we get
x( 2x2 + 5x - 3 ) = 0
2x2 + 5x - 3 = 0
using splitting middle term method and get
2x2 + 6x - x - 3 = 0
2x ( x + 3 ) - 1 ( x + 3 )
(2x - 1 ) ( x + 3 ) = 0
So here zeros will be
x = 0 , - 3 ,
here we can see that our polynomial have different zeros .
b ) 0 may be a zero of a polynomial.
yes that can be posible , lets take an example :
we have a polynomial
f( x ) = 2x3 + 5x2 - 3x
So we check for x = 0
f( x ) = 2 ( 0 ) + 5 ( 0 ) - 3 ( 0 )
f(x) = 0
Hence 0 is a zeros of polynomial 2x3 + 5x2 - 3x
c ) Every linear polynomial has one and only one zero.
We know linear polynomial As y = a x + b
So,
Let a linear polynomial y = 2x - 4
Then
2x - 4 = 0
x = 2 ( Single root )
Hence
We can say that statement is true " Every linear polynomial has one and only one zero. "
d ) A polynomial can have more than one zero.
Let we have a polynomial x2 -2x - 3 .
We can find out its zeros by using splitting the middle term method As :
x2 - 3x + x - 3
x(x - 3 ) + 1 ( x - 3 )
(x + 1 ) ( x - 3 )
So,
Zeros are -1 and 3
So we can say that statement is true " A polynomial can have more than one zero. "
a ) A zero of a polynomial need not be 0.
that statement is right , because polynomial can have different zeros that can be zero or apart from that .
Example : we have a polynomial
f( x ) = 2x3 + 5x2 - 3x
we can write it As: x( 2x2 + 5x - 3 )
Equate equal to zero , we get
x( 2x2 + 5x - 3 ) = 0
2x2 + 5x - 3 = 0
using splitting middle term method and get
2x2 + 6x - x - 3 = 0
2x ( x + 3 ) - 1 ( x + 3 )
(2x - 1 ) ( x + 3 ) = 0
So here zeros will be
x = 0 , - 3 ,
here we can see that our polynomial have different zeros .
b ) 0 may be a zero of a polynomial.
yes that can be posible , lets take an example :
we have a polynomial
f( x ) = 2x3 + 5x2 - 3x
So we check for x = 0
f( x ) = 2 ( 0 ) + 5 ( 0 ) - 3 ( 0 )
f(x) = 0
Hence 0 is a zeros of polynomial 2x3 + 5x2 - 3x
c ) Every linear polynomial has one and only one zero.
We know linear polynomial As y = a x + b
So,
Let a linear polynomial y = 2x - 4
Then
2x - 4 = 0
x = 2 ( Single root )
Hence
We can say that statement is true " Every linear polynomial has one and only one zero. "
d ) A polynomial can have more than one zero.
Let we have a polynomial x2 -2x - 3 .
We can find out its zeros by using splitting the middle term method As :
x2 - 3x + x - 3
x(x - 3 ) + 1 ( x - 3 )
(x + 1 ) ( x - 3 )
So,
Zeros are -1 and 3
So we can say that statement is true " A polynomial can have more than one zero. "