Question

find the remainder when 4 x cube minus 12 x square + 14 x minus 3 is divided by 2x - 1 using long division method ​

Answer 1

$f(x) = 2x - 1 = 0 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: x = \frac{1}{2}$

Now,

$p(x) = {4x}^{3} - {12x}^{2} + 14x - 3 \\ \\ p( \frac{1}{2}) = {4( \frac{1}{2} )}^{2} - 12 {( \frac{1}{2} )}^{2} + 14( \frac{1}{2} ) - 3 \\ \\ \: \: \: \: \: \: \: \: \: \: = 4( \frac{1}{16} ) - 12( \frac{1}{4} ) + 7 - 3\\ \\ \: \: \: \: \: \: \: \: \: \: = \frac{1}{4} - 3 + 7 - 3 \\ \\ \: \: \: \: \: \: \: \: \: \: = \frac{1 - 12 + 28 - 12}{4} \\ \\ \: \: \: \: \: \: \: \: \: \: = \frac{29 - 24}{4} = \frac{5}{4} = 1.25$