Question

Assertion : The sum and product of the zeros of a quadratic polynomial are 1
4
and −1
4
respectively. Then

the quadratic polynomial is 4x

2 + x + 1.

Reason: The Quadratic polynomial whose sum and product of the zeros are given is
x
2 − (sum of zeros). x + product of zeros.

Answer 1

Answer:

Zeros of Quadratic Polynomial

The zeros of a polynomial are those values of the variable for which the polynomial as a whole has zero value. For example, consider the linear polynomial: p(x) = x+2. The zero of the polynomial is -2 because when x = -2, p(x)=0. Consider a quadratic polynomial: p(x)= x2−9. The zeros of the polynomial are ±3. When x = ±3, p(x)=32−9=0.

Note that when the coefficient of all the terms and the constant of the polynomial is 0, the polynomial is called a zero polynomial and is denoted by 0.

A polynomial of degree n will have n number of zeros or roots. Thus, a quadratic polynomial can have at most two zeros, whereas a cubic polynomial can have at most 3 zeros.

Answer 2

Both assertion and reason are true and the reason is the correct explanation of the assertion.

Let the roots of the equation are p and q

and we are given that sum of the roots of the quadratic equation will be p+q = -1/4

product of the roots of the quadratic equation will be

p×q= 1/4

and we know that

x^2 - ( sum of roots )x + ( product of roots)

x^2 - ( p+q) + ( p×q )

x^2 - ( -1/4)x + 1/4

1/4( 4x^2 + x + 1)

Therefore, quadratic polynomial will be 4x^2+x+1 = 0.

Both assertion and reasoning is correct.