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The number of polynomials having zeroes ...

The number of polynomials having zeroes as -2 and 5 is

A

1

B

2

C

3

D

more than 3

Text Solution

Verified by Experts

The correct Answer is:
D
Let `p(x) = ax^(2) +bx +c` be the required polynomial whose zeroes are -2 and 5.
`:.` Sum of zeroes `= (-b)/(a)`
`rArr (-b)/(a) =- 2 +5 =(3)/(1) = (-(3))/(1)`
and product of zeroes `=(c)/(a)`
`rArr (c)/(a) =- 2 xx 5 =(-10)/(1)`
From Eqs. (i) and (ii),
`a = 1, b =- 3` and `c =- 10`
`:. p(x) = ax^(2) +bx +c = 1x^(2) - 3x - 10`
`= x^(2) - 3x - 10`
But we know that, if we multiply/divide any polynomial by any arbitary constant. Then, the zeroes of polynomial never change.
`:. p(x) = kx^(2) - 3kx - 10k` [where, k is a real number]
`rArr p(x) =(x^(2))/(k)-(3)/(k)x -(10)/(k)`, [where,k is a non-zero real number]
Hence, the required number of polynomials are infinite i.e., more than 3.
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