Question

What must be added to 4x⁴ + 2x³ - 2x² + x -1 so that the resulting polynomial is divisible by x² +2x - 3 ?

Answer 1

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).

HOPE THIS WILL HELP YOU...

Answer 2

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).