Given
Let f( x) = x³ + 3x² + 1
g( x) = x + 1
Find the zero of g( x)
To find the zero of polynomial equate it to zero
x + 1 = 0
x = - 1
By Remainder theorem
If f( x) is divided by g( x) then the remainder is f( zero of g( x) )
Therefore by the theorem,
f( - 1 ) is the remainder
f( x) = x³ + 3x² + 1
f( - 1 ) = ( - 1 )³ + 3( - 1 )² + 1
f( - 1 ) = - 1 + 3( 1 ) + 1
f( - 1 ) = 3
Therefore the remainder when x³ + 3x² + 1 divided by x + 1 is 3.