Question

The number of polynomials having zeros as -2 and 5?
A) 0
B) 2
C) 3
D) More than three

Answer 1

Hint: Use the standard equation of a polynomial, then the property of the sum of its zeroes and product of zeros. Equate the equations formed by these properties, you will get a polynomial and using this polynomial proceed further.

Complete step-by-step answer:

Let \[p(x) = a{x^2} + bx + c\] is the required polynomial whose zeroes are -2 and 5.
∴ Sum of zeroes = \[\dfrac{{ - b}}{a}\]
\[ \Rightarrow \dfrac{{ - b}}{a} = - 2 + 5 = \dfrac{3}{1} = \dfrac{{ - \left( { - 3} \right)}}{1}\]………..(1)
And product of zeroes =\[\dfrac{c}{a}\]
\[ \Rightarrow \dfrac{c}{a} = - 2 \times 5 = \dfrac{{ - 10}}{1}\]…………….(2)
From equations (1) and (2)
a = 1, b = -3 and c = -10
∴\[p(x) = a{x^2} + bx + c = 3{x^2} - 3x - 10\]
But we know that, if we multiply or divide any polynomial by any arbitrary constant. Then, the zeroes of polynomial never change
∴\[p(x) = k{x^2} - 3kx - 10k\], where k is the real number
\[ \Rightarrow p(x) = \dfrac{{{x^k}}}{k} - \dfrac{3}{k}x - \dfrac{{10}}{k}\], where k is a non-zero real number
Hence, the required polynomials are infinite i.e. more than three
∴ The correct option is ‘D’.

Note: A polynomial is an expression consisting of variables (also called indeterminate) and coefficients, that involve only operations of addition, subtraction, multiplication, and non-negative integer exponents of the variable.