Find a quadratic polynomial Px ax2 bx c where the coeffic

Find a quadratic polynomial P(x) = ax^2 + bx + c, where the coefficients a, .b, c belong to {0, 1, 2, 3, 4}, such that the following all hold: p(1) equiv 1 (mod 5), p(2) equiv 2 (mod 5), p(3) equiv 4 (mod 5),

Solution

P(1)=a+b+c=1 mod 5

P(2)=4a+2b+c=2=3a+b+a+b+c=3a+b+1=2 mod 5

So, 3a+b=1 mod 5

P(3)=9a+3b+c=4a+3b+c=3a+2b+a+b+c=3a+2b+1=4 mod 5

3a+2b=3 mod 5 and 3a+b=1 mod 5

He,ce b=2 mod 5

But, b belongs to {0,1,2,3,4}

Hence, b=2

3a+b=1 mod 5

3a+2=1 mod 5

3a=-1 mod 5

3*2a=-2 mod 5

a=-2=3 mod 5

a=3 mod 5

Hence, a=3

a+b+c=1 mod 5

3+2+c=5+c=c=1 mod 5

Hence, c=1

So

P(x)=3x^2+2x+1