Question

Find the remainder when x^3 + 3x^2 + 3x + 1 is divided by x -1/2.

Answer 1

$\huge \: answer : \\ \\ by \: remainder \: \: theorem \: \\ \\ g(x) = x - \frac{1}{2} \\ \\ p(x) = {x}^{3} + {3x}^{2} + 3x + 1 \\ \\ g(x) = 0 \\ \\ x - \frac{1}{2} = 0 \\ \\ x = \frac{1}{2} \\ \\ now \\ \\ p( \frac{1}{2} ) = ( { \frac{1}{2} })^{3} + 3( \frac{1}{2} ) ^{ 2 } + 3( \frac{1}{2} ) + 1 \\ \\ = \frac{1}{8} + 3( \frac{1}{4} ) + \frac{3}{2} + 1 \\ \\ = \frac{1}{8} + \frac{3}{4} + \frac{3}{2} + 1 \\ \\ = \frac{1}{8} + \frac{6}{8} + \frac{12}{8} + \frac{8}{8} \\ \\ = \frac{1 + 6 + 12}{8} + 1 \\ \\ = \frac{19}{8} + 1 \\ \\ = \frac{27}{8}$