Question

If f(x)=2x⁴-5x²+x²+3x-2 is divided by g(x) the quotient q(x)=2x²-5x+3 and r(x)=-2x+1. Find g(x)

Answer 1

$divident = divisor \times quotient + remainder$
$f(x) =g(x) \times q(x) + r(x) \\ 2 {x}^{4} - 5 {x}^{3} \ + {x}^{2} + 3x - 2 = \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ g(x) \times( 2 {x}^{2} - 5x + 3) + ( - 2x + 1) \\ 2 {x}^{4} - 5 {x}^{3} + {x}^{2} + 5x - 3 = g(x) \times (2 {x}^{2} - 5x + 3) \\ g(x) = \frac{2 {x}^{4} - 5 {x}^{3} + {x}^{2} + 5x - 3 }{2 {x}^{2} - 5x + 3} \\ 2 {x}^{2} - 5x + 3)2 {x}^{4} - 5 {x}^{3} + {x}^{2} + 5x - 3( {x}^{2} - 1 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 2 {x}^{4} - 5 {x}^{3} + 3 {x}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - \: \: \: \: \: \: \: + \: \: \: \: \: \: \: - \\ - 2 {x}^{2} + 5x - 3 \\. - 2 {x}^{2} + 5x - 3 \\ + \: \: \: \: \: \: - \: \: \: \: \: \: + \\ - - - - - - - - - - - - - - - - - - \\ 0$
$g(x) = {x}^{2} - 1$