the polynomial `p(x)=x^(4)-2x^(3)+3x^(2)-ax+3a -7` when divided by x+1 leaves the remainder 19. find the values of A , also find the remainder when p(x) is divided by x+2.

the polynomial `p(x)=x^(4)-2x^(3)+3x^(2)-ax+3a -7` when divided by x+1

1.	the polynomial `p(x)=x^(4)-2x^(3)+3x^(2)-ax+3a -7` when divided by x+1 leaves the remainder 19. find the values of A , also find the remainder when p(x) is divided by x+2.
Answer» Given `p(x)=x^(4)-2x^(3)+3x^(2)-ax+3a-7` when we divide p(x) by x+1 ,then we get the the reamainder p(-1) Now , `P(-1)=(-1)^(4)-2(-1)^(3)+3(-1)^(3)+3(-1)^(2)-a(-1)+3a-7` `=1+2+3+a+3a-7=4a-1` According to the question `p(-1) =19` `implies 4a-1=19` `implies 4a-1=19` ` implies 4a=20` `therefore a=5` `therefore ` Required polynomial `=x^(4)-2x^(3)+3x^(2)-5x+3(5)-7` `=x^(4) -2x^(3) +3x^(2)-5x+15-7` `=x^(4)-2x^(3) +3x^(2)-5x+8` when we divide `p(x) ` by x+2 then we get the remainder p(-2), Now , `P(-2)=(-2)^(4)-2(-2)^(3)+3(-2)^(3) +3(-2)^(2)-5(-2)+8` `=16+16+12+10+8=62` hence ,the value of a is 5 and remainder is 62 .