Question

1)Find a second degree polynomial p(x) such that p(1) =0 and p(-2) =0.

2)Find a second degree polynomial p(x) such that p(1+√3)=0 and p(1-√3)=0.

3)Find a third degree polynomial p(x) such that p(1)=0,p(√2)=0 and p(-√2)=0.

Answer 1

HOPE THIS HELPS YOU MISTER

Answer 2

1) Given 1, -2 are zeroes of polynomial.
    Therefore, x =1 and x = -2 
     So, factors are (x-1) and (x+2)
      Therefore polynomial = (x-1)(x+2)
                                         = x(x+2) -1(x+2)
                                         = x² + 2x -x -2
      So, polynomial is        = x² + x -2
2) Given (1+√3) and (1-√3) are zeroes of polynomial
     Therefore, x= 1+√3 and x= 1-√3
     So, factors are (x-√3-1) and (x+√3-1)
   Therefore polynomial = (x-√3-1)(x+√3-1)
                                      = x(x+√3-1) -√3(x+√3-1) -1(x+√3-1)
                                      = x²+√3x-x -√3x-3+√3 -x-√3+1
    So, polynomial is       = x²-2x-2
3) Given 1, √2 and -√2 are zeros of polynomial.
     Therefore x=1, x=√2, x= -√2
     So, factors are (x-1) , (x-√2) and (x+√2)
     Therefore polynomial = (x-1)(x-√2)(x+√2)
                                       =  (x-1){x²-(√2)²}     [using (a+b)(a-b) = a²-b²]
                                       =  (x-1)(x²-2)
                                       = x(x²-2) -1(x²-2)
                                       = x³-2x -x²+2
       So. polynomial is    = x³-x²-2x+2