Question

Find the quadratic polynomial in ‘x’ which when divided by (x – 1), (x – 2) and (x – 3) leaves the remainder of 11, 22 and

37 respectively.​

Answer 1

We know that the remainder when f(x) is divided by x−α is f(α). If we say that

f(x)=ax2+bx+c

then we have

a×12+b×1+c =11 a+b+c =11 a×22+b×2+c =22 4a+2b+c =22 a×32+b×3+c =37 9a+3b+c =37

We then have 3 equations in 3 unknowns so we can solve these for a,b,c: (2)−(1):

3a+b=11

(3)−(2):

5a+b=15

(5)−(4):

2a=4a=2

From (4):

3×2+b=11b=5

From (1):

2+5+c=11c=4

And therefore, our quadratic is:

f(x)=2x2+5x+4