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Zeroes of a Polynomial

Now consider a polynomial,

P(x) =

Put, x = 1, in P(x),

We get, 4(1) + 5(1) + 6(1) + 24 = 4 + 5 + 6 + 24 = 39

Therefore, the value of P(x) at x = 1 is 39.

 

Now consider a polynomial,

R(x) =

Put, x=2 in R(x)

R(x) = 3(4) + 3(2) +

       = 12 + 6 +

       = 18 +

 

Example : We have seen that the value to the polynomial is not zero. Now we are going to see an example, in such a way that we get the value of the polynomial is zero.

Consider a polynomial, p(x) = at x = 1,

                                      = 1-2+1

                                      = 0

The value of p(x) at x = 1 is 0

Then we say that x = 1 is the zero of the polynomial or root of the polynomial.

 

In general, we say that a number c such that p(c)=0 in a polynomial p(x). then c is the zero of the polynomial or the root of the polynomial.

If the polynomial takes the value zero, then the value x is called the zero of the polynomial. Such a number is called root of a polynomial.

For a non zero constant polynomial, say 9 it has no zero since by replacing x in gives the value 9.

Every real number is a zero of a zero polynomial.

 

Example : 1 Check Whether -4 and 4 are the roots of the polynomial

Let p(x) =

p(4)= 4 x 4 – 16 = 0

Therefore, 4 is the root of P(x)

 

P (-4) = 4 x (-4) -16 = -16 – 16 = -32

Therefore, -4 is not a root of

p(x)

 

Example : 2 Check whether -3 and 4 are the roots of the polynomial

Let

p(x) =

    p(-3 ) = -3x -3 + -(-3) -12

           = 9 +3 -12 = 0

Therefore, -3 is a root of the polynomial

p(4) = 4x4 4-12

       = 16-16 = 0

Therefore, 4 is also a root of the polynomial

Therefore -3 and 4 are the roots of the polynomial

From the above examples we infer that,

Every linear polynomial has one and only one root.

Polynomial has more than one root.

 





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