(Q9) Check in which case the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial

x2+3x+1 , 3x4+5x3-7x2+2x+2

Given first polynomial is x2+3x+1

Second polynomial is 3x4+5x3-7x2+2x+2

Let us divide by 3x4+5x3-7x2+2x+2 by x2+3x+1 we get

p(x)=x4-3x2+4x+5

g(x)=x2+1-x

Here, dividend already in the standard form and the divisor is not in the standard form. it can be written as x2-x+1

So, we have

x2-x+
x2+3x+1) x4+x3-x2+x+
x4+x3+x2
-    -   -
-x3-x2+x+
-x3-x2-x
+   +   +
x2+x+
x2+x+
-   -   -

Since the remainder is , therefore 2+x+ is a factor of 3x4+5x3-7x2+2x+2