Question

what must be added to f(x)= 4x^4 + 2x^3 - 2x^2 + x - 1, so that the resulting polynomial is divisible by g(x) =x^2+ 2x - 3 (please show all calculations)

Answer 1

Answer:

The number which should be added to the polynomial is

Step-by-step explanation:

It is given that p[ x ] =4x⁴+2x³-2x²+x-1 + x to get resultant becomes g[ x ] =x²+2x-3

We must use remainder theorem to get the resultant

g[ x ] =x²+2x-3

x²+2x-3= 0

x²+3x-x-3=0

[ x²+3x ] - [ x+3 ] =0

x [ x+3] - 1 [ x+3 ] =0

[ x-1 ] [ x+3 ] =0

x = 1 or -3

f[ x ] =4x⁴+2x³-2x²+x-1 when x is 1

4[1⁴]+2[1³]-2[1²]+1-1

4[1]+2[1]-2[1]+0

4+2-2

4 should be added

f[ x ] = 4x⁴+2x³-2x²+x-1 when x is -3

4[-3]⁴+2[-3]³-2[-3]²+[-3]-1

4[81]+2[-27]-[9]-3-1

324-54-9-3-1

257 should be added