Question

what should be added to 14x³+29x²-37x+51 to get 2x³-10x²+2x-23​

Answer 1

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).