Question

What must be added to the polynomial f(x) = x⁴ + 2x³ – 2x² + x – 1 so that the resulting polynomial is exactly divisible by g(x) = x² + 2x – 3 ?

Answer 1

Answer:
Step-by-step explanation:
By division algorithm:
Dividend, f(x) =  Divisor g(x) × quotient q(x)+ remainder r(x)
f(x) - r(x) = g(x) × q(x)
f(x) + {- r(x)} = g(x) × q(x)
If we add - r(x) to f(x)  then the resulting polynomial is divisible by g(x). Now we  find the remainder when f(x) is divided by g(x).
x² +2x - 3)4x⁴ + 2x³ - 2x²  +  x -1(4x²-6x+22
  4x⁴ + 8x³- 12x²
   (-)    (-)     (+)
 -----------------------------
   -6x³ +10x²+ x-1
 -6x³ -12x²+18x
   (+)    (+)    (-)
 -------------------------------
 22x² -17x -1
 22x² + 44x -66
  (-)     (-)     (+)   
------------------------------  
-61x +65
r(x) =  -61x +65
Hence, we should add -r(x) =  61x - 65 to f(x) so that the resulting polynomial is divisible by g(x).

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