Question

Quadratic polynomials is exactly divisible by (x-1) (x+2) and leaves a remainder 4 after division by (x+3) then that polynomial is-(with proper steps plz)

Answer 1

Thne polynomial is quadratic, means it has 2 factors.
Question says (x-1) and (x+2) are exactly divisible, means they are it's factors.
That's it!
The polynomial is,

$(x - 1)(x + 2) = {x}^{2} +2x - x - 2 \\ = {x}^{2} + x - 2$
Note: I don't why the question says about (x+3) and reminder 4. May to confuse. But if you divide the above answer with (x+3), you will get a reminder = 4
Hope this helps

Answer 2

Answer:

since quadratic polynomial =ax^ 2+ bx+ c (1)

p(X) =ax^2 + bx+ c

g(X) = X+1

x+1 =0

x= -1

p(-1) = a (-1)^2+b(-1) + c

= a+ -b + c. (1)

p(X) = ax^2+bx + c

g(X) = X+ 2

x+2 =0

x=-2

p(-2) = a(-2)^2+b(-2) + c

= 4a - 2b + c. (2)

p(X) = ax^2 +bx+ c

g(X) = X+3

x+3=0

x=-3

p(-3) = a (-3)^2 + b(-3) +c

= 9a -3b+ c. (3)

from (2)- (1)

-3a + b= 0

b= 3a

put value of b=3a in equation 3

since remainder =4

therefore 9a -9a +c = 4

c=4

a-b+4=0. (4)

4a-2b+4= 0 (5)

multiple equation 4 by 2

2a-2b+ 8

4a-2b+4

-. +. -

-2a. 4

a= 2

when a=2 , b=3a, so b=6

so quadratic polynomial = ax^2+bx+c

= 2x^2+6x+ 4

I hope it help you