Question

find the quadratic polynomial when divided by x,x-1,x-2 leaves remainder 1 , 2 and 9 respectively

Answer 1

The quadratic polynomial  is P(x) = 3 x² - 2 x + 1

Step-by-step explanation:

Given that

• Quadratic polynomial are

        P(x) = a x² + b x +c     ...1)

• When it  divide by (x-a) leaves remainder b.

        Put

        x-a = 0    ⇒x =a

        From remainder theorem from equation 1)

        P(a) = b

• First case when divide by x leaves remainder 1.

        x = 0

        So by remainder theorem in equation 1)

        P(0) = 1

        a× 0² + b× 0 +c =1

        c =1

• Second case when divide by (x-1) leaves remainder 2.

        x-1 = 0     ⇒x =1

        So by remainder theorem in equation 1)

        P(1) = 2

        a× 1² + b× 1 +c =2         (where c = 1)

        a +b = 1             ...2)

• Third case when divide by (x-2) leaves remainder 9.

        x-2 = 0     ⇒x =2

        So by remainder theorem in equation 1)

        P(2) = 9

        a× 2² + b× 2 +c =9         (where c = 1)

        4a +2b = 8

        means

       2a + b = 4          ...3)

• On solving equation 2) and equation 3) , we get

        a =3    and b = - 2

• On putting value of a,b and c in equation 1)

        P(x) = 3 x² - 2 x + 1