Multiple choice

Given $ax^2+bx+c$ is a quadratic polynomial in x and leaves remainders 6, 11 and 18 respectively when divided by $(x+1), (x+2) and (x+3)$. Find the value of $a+b+c.$

  1. $1$
  2. $2$
  3. $3$
  4. $4$
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B Correct answer
Explanation

By Remainder Theorem, P(-1)=6, P(-2)=11, P(-3)=18. Let P(x) = ax^2 + bx + c. P(-1)=a-b+c=6, P(-2)=4a-2b+c=11, P(-3)=9a-3b+c=18. Subtracting equations: (4a-2b+c)-(a-b+c) = 3a-b = 5. (9a-3b+c)-(4a-2b+c) = 5a-b = 7. Subtracting these: 2a=2 => a=1. Then 3(1)-b=5 => b=-2. Then 1-(-2)+c=6 => c=3. a+b+c = 1-2+3 = 2.

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