Question

The number of polynomials having zeros as 2 and 5 is:
A. 1
B. 2
C. 3
D. >3

Answer 1

Hint:In this question, we will use a general form of polynomial with given values of zeroes to find the number of polynomials having zeros 2 and 5.

Complete step-by-step answer:
A polynomial which has 2 as its root or zero, will have a factor which when equated to zero will give the value of the variable to be 2.
Let the variable of polynomials be x. Then, for the value of x to be 2, we have, $x=2$. Subtracting 2 from both side of the equation, we get,
$x-2=0$
So, $\left( x-2 \right)$ will be a factor of required polynomials. Also, 5 is also a zero of this polynomial.
So, this polynomial will also have a factor which when compared to zero gives value 5 of the variable. So, for x to be 5, we have, $x=5$ . Subtracting 5 from both side of the equation we have,
$x-5=0$
So, $x-5$ Will be a factor of required polynomials.
Also, for any number of these factors, zeros of polynomials will still be 2 and 5.
Let us consider a polynomial, number of factors $x-2$ be n and number of factors of $x-5$ be m.
And, if we multiply these factors with a Scalar, Value of the polynomial won’t change. Hence the required polynomial is $K{{\left( x-2 \right)}^{n}}{{\left( x-5 \right)}^{m}}$ , where K is scalar.
Here the value of n and m can be any natural number. And for each value of each n and m we will have different polynomials. Therefore, this can be an infinite number of polynomials with zeros 2 and 5.
Hence the correct answer is option (d).

Note: Zeros of a polynomial can be defined as the points where the polynomial becomes zero on the whole. A polynomial having value zero (0) is called zero polynomial. The degree of a polynomial is the highest power of the variable x. A polynomial of degree 1 is known as a linear polynomial.In this type of question, whenever we are not informed how many times the same zero can appear, then the number of polynomials will be infinite.