Question

The number of polynomials having zeroes as 4 and 7 is
a)2
b)3
c)4
d)more than 4

Answer 1

Answer:

d

Step-by-step explanation:

answer is more than 4

Answer 2

The number of polynomials having zeroes as 4 and 7 is more than 4

Given :

The zeroes of a polynomial are 4 and 7

To find :

The number of polynomials having zeroes as 4 and 7 is

a) 2

b) 3

c) 4

d) more than 4

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 4 and 7

Sum of zeroes = 4 + 7 = 11

Product of the zeroes = 4 × 7 = 28

Step 2 of 2 :

Find the number of polynomials

The polynomial is of the form

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

$\displaystyle \sf =k( {x}^{2} -11x + 28)$

Where k is a non zero real number

Since k can be any non zero real number

So there are infinite number of polynomials having zeroes 4 and 7

Hence the correct option is d) more than 4

Method : 2

Solution :

Step 1 of 2 :

Write down the given zeroes

Here it is given that the zeroes of a polynomial are 4 and 7

Step 2 of 2 :

Find number of polynomials

Let 4 and 7 are zeroes of the polynomial of multiplicity m and n respectively

Then the polynomial is of the form

$\displaystyle \sf{ k {(x - 4)}^{m} {(x - 7)}^{n} }$

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

So there are infinite number of polynomials having zeroes 4 and 7

Hence the correct option is d) more than 4

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