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

Explanation

We know that if p(x) is divided by x+a, then the remainder =p(-a).
Now, `p(x)=x^(4)-2x^(3)+3x^(2)-ax+3a-7` is divided by x+1, then the remainder =p(-1)
Now, `p(-1)=(-1)^(4)-2(-1)^(3)+3(-1)^(2)-a(-1)+3a-7`
=1-2(-1)+3(1)+a+3a-7=1+2+3+4a-7
=-1+4a
Also, remainder =19
`therefore -1+4a=19`
`implies 4a=20 implies a=20+4=5`
Again, when p(x) is divided by x+2, then
remainder `=p(-2)=(-2)^(4)-2(-2)^(3)+3(-2)^(2)-a(-2)+3a-7`
=16+16+12+2a+3a-7
=37+5a=37+5(5)=37+25=62