Question

find the remainder when x cube + 3 X square + 3x + 1 is divided by x-1/2​

Answer 1

Let's take:

$p(x) = {x}^{3} + 3 {x}^{2} + 3x + 1$

$g(x) = x - \frac{1}{2}$

Using remainder theorem to find the remainder when p(x) is divided by g(x).

Remainder theorem: If a polynomial p(x) is divided by another polynomial (x-a) then the remainder will be p(a).

Finding the value of x in g(x):

$x = \frac{1}{2}$

Substituting the value of x in p(x):

${( \frac{1}{2}) }^{3} + 3( { \frac{1}{2}) }^{2} + 3( \frac{1}{2} ) + 1 \\ = \frac{1}{8} + \frac{3}{4} + \frac{3}{2} + 1 \\ = \frac{1 + 6 + 12 + 8}{8} \\ = \frac{27}{8}$

Therefore, the remainder when p(x) is divided by g(x) is 27/8.

Answer 2

Answer

$(p(x)=x 3+3x 2 +3x+1)$

$(g(x) = x - (x)=x− 21$

x = x= 21

21 ) +3( 21 ) 2 +3( 21 )+1= 81 + 43 + 23

+1= 81+6+12+8

27/8