Question

What must be subtracted from the polynomial $f(x)=x^{4}+ 2x^{3}-13x^{2}-12x+21$ so that the resulting polynomial is exactly divisible by $x^{2}+ 4x+3?$

Answer 1

A polynomial is exactly divisible by another polynomial, if remainder is zero. So , here find the remainder and subtract  remainder from f(x) , so that the resulting polynomial is divisible by g(x).

Given : f(x) = x⁴ + 2x³ - 13x² - 12x +21

and g(x) = x² - 4x + 3

Now in Dividing f(x) by g(x) , we get the following division process.  

DIVISION PROCESS  is in the attachment.

Here, Remainder is 2x -3 . Now the polynomial  f(x) = x⁴ + 2x³ - 13x² - 12x +21 will be exactly divisible by g(x) = x² - 4x + 3, when reminder is zero.  So to make the remainder 0 , 2x -3 is to be subtracted from f(x) .  

Hence, if we subtract r(x) =  2x -3 from  f(x) , then it will be divisible by g(x) =x² - 4x + 3

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

Out of lazyness, i’m just gonna only write the coefficients and subtraction results:

1,2,-13,-12,21 : 0,0,1,-4,3 = 0,0,1,6,8

0,6,-16,-12,21

0,0,8,-30,21

0,0,0,2,-3

So the answer of the division is x^2+6x+8+(2x-3)/(x^2–4x+3)

If you want the last term to be gone, subtract by 2x-3 and get as initial function

1,2,-13,-14,24