# 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.

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Now consider a polynomial,

*R(x) = *

Put*, x=*2 in R(*x*)

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

Â Â Â Â Â Â = 12 + 6 +

Â Â Â Â Â Â = 18 +

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**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,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 1-2+1

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 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.

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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.

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

Let *p*(*x*) =

p(4)= 4 x 4 â€“ 16 = 0

Therefore, 4 is the root of *P(x)*

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*P*(-4) = 4 x (-4) -16 = -16 â€“ 16 = -32

Therefore, -4 is not a root of

*p(x) *

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

Let

*p(x)*=

*Â Â Â p*(*-*3* *)* = -*3x -3 + -(-3) -12

Â Â Â Â Â Â Â Â Â Â = 9 +3 -12 = 0

Therefore, -3 is a root of the polynomial

p(4) = 4x4 4-12

Â Â Â Â Â Â = 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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