For a quadratic polynomial f(x), f(0) = 0 and f(1) = 1, and sum of the roots is 2. Find the equation.

Concept Used:

For a quadratic polynomial,

P(x) = ax² + bx + c

Sum of the roots

A product of roots

Explanation:

f(x) = ax² + bx + c

f(0) = a.0 + b.0 + c = 0

C = 0

Therefore,

Polynomial becomes,

f(x) = ax² + bx

And,

f(1) = a.1 + b.1 = 1

a + b = 1

b = 1 – a

Now, the equation becomes,

f(x) = ax² + (1 – a)x

Sum of the roots

2a = a – 1

a = –1

So, the final polynomial is,

f(x) = – x² + (1 + 1)x

Hence, the final quadratic polynomial is f(x) = – x² + 2x