Question

if on dividing the polynomial f (x) = 2x4 - 3x2 - ax plus b (x - 1) and (x plus 1) the remainder are -3 and 7 respectively, then find the remainder when f (x) is divided by (x -2)

Answer 1

Hi,

$\left[\begin{array}{l|ccccc}&x^4&x^3&x^2&x&1\\&2&0&-3&-a&b\\x=1&&2&2&-1&-a-1\\&2&2&-1&-a-1&b-a-1\\\end{array}\right] \\\\\Rightarrow\ b-a-1=-3\\\Rightarrow\ -a+b=-2\\\\\left[\begin{array}{l|ccccc}&x^4&x^3&x^2&x&1\\&2&0&-3&-a&b\\x=-1&&-2&2&1&a-1\\&2&-2&-1&-a+1&b+a-1\\\end{array}\right] \\\\\Rightarrow\ b+a-1=7\\\Rightarrow\ a+b=8\\\\\left \{ {{-a+b=-2} \atop {a+b=8}} \right.\\\\\left \{ {{a=5} \atop {b=3}} \right.$

$\left[\begin{array}{l|ccccc}&x^4&x^3&x^2&x&1\\&2&0&-3&-5&3\\x=2&&4&8&10&10\\&2&4&5&5&13\\\end{array}\right] \\\\\Rightarrow\ b-a-1=-3\\\Rightarrow\ -a+b=-2$

remainder is 13.