Question

The number of polynomials having zeroes –2 and 3 is

Answer 1

Hence, the number of polynomials are infinite.

Answer 2

The number of polynomials having zeroes – 2 and 3 is infinite

Given :

The zeroes of a polynomial are – 2 and 3

To find :

The number of polynomials having zeroes – 2 and 3

Method : 1

Concept :

If the Sum of zeroes and Product of the zeroes of a quadratic polynomial is given then the quadratic polynomial is

$\sf{k[ {x}^{2} -(Sum \: of \: the \: zeroes )x + Product \: of \: the \: zeroes] }$

Where k is a non zero real number

Solution :

Step 1 of 2 :

Find Sum of zeroes and Product of the zeroes

Here it is given that zeroes of a polynomial are – 2 and 3

Sum of zeroes = – 2 + 3 = 1

Product of the zeroes = ( – 2) × 3 = – 6

Step 2 of 2 :

Find the number of polynomials

The polynomial is of the form

$\displaystyle \sf = k[{x}^{2} -(Sum \: of \: the \: zeroes )x + Product \: of \: the \: zeroes]$

$\displaystyle \sf =k( {x}^{2} -x - 6)$

Wheee k is a non zero real number

Since k can be any non zero real number

Hence there are infinite number of polynomials having zeroes – 2 and 3

Method : 2

Solution :

Step 1 of 2 :

Write down the given zeroes

Here it is given that the zeroes of a polynomial are – 2 and 3

Step 2 of 2 :

Find number of polynomials

Let – 2 and 3 are zeroes of the polynomial of multiplicity m and n respectively

Then the polynomial is of the form

$\displaystyle \sf{ k {(x + 2)}^{m} {(x - 3)}^{n} }$

Where m and n are natural numbers

Also k is a non zero real number

Since m, n are natural numbers and k is a non zero real number

Hence there are infinite number of polynomials having zeroes – 2 and 3

Final Answer :

The number of polynomials having zeroes – 2 and 3 is infinite

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