what no . should be subtracted from x ^ 3 + 4 x 2 − 7 x + 12 x ^ 3 + 4 x ^ 2 − 7 x + 12 , if it is to be perfectly divisible by x + 3 x + 3 ?
according to remainder theorem when dfracf ( x ) x + adfracf ( x ) x + a , then the remainder is f ( − a ) f ( − a ) . in this case , as x + 3 x + 3 divides x 3 + 4 x 2 − 7 x + 12 – kx 3 + 4 x 2 − 7 x + 12 – k perfectly ( kk being the number to be subtracted ) , the remainder is 0 when the value of xx is substituted by - 3 . i . e . , ( − 3 ) 3 + 4 ( − 3 ) 2 − 7 ( − 3 ) + 12 − k = 0 ( − 3 ) 3 + 4 ( − 3 ) 2 − 7 ( − 3 ) + 12 − k = 0 or − 27 + 36 + 21 + 12 = k − 27 + 36 + 21 + 12 = k or k = k = 42 b
add(subtract(add(power(negate(3), 3), multiply(4, power(negate(3), 2))), multiply(7, negate(3))), 12)