Question

find the remainder when 2 x cube minus 5 x square + X + 2 is divided by X + 1

Answer 1

2x³ - 5 x² + x +2 divided by x+1

Solution =

Consider x+1 = 0

x = -1

Now,

Put x=-1

=2(-1)³ - 5 (-1)² + (-1) +2

=2×-1-5(1)-1+2

=-2×-5-1+2

= -6

The remainder is 5 .

Answer 2

Let 2x³ - 5x² + x + 2 = f( x )




Since x + 1 = x - ( - 1 ), by cor. 1 to remainder theorem :


$\text{Remainder }= f( - 1 ) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 2( - 1) {}^{3} - 5 {( - 1)}^{2} + ( - 1) + 2 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = - 2 - 5 - 1 + 2 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = - 2 + 2 - 5 - 1 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = -6$




Therefore,
On dividing ( 2x³ - 5x² + x + 2 ) by ( x + 1 ) , we get -6 as remainder.