# Find the values of k and p so that the expression -2x^3-kx^2+7x+p can be divided by 2x^2+5x+3 without leavng any remainder.?

Jul 13, 2016

$\left\{k = 1 , p = 6\right\}$

#### Explanation:

Calling

$f \left(x\right) = - 2 {x}^{3} - k {x}^{2} + 7 x + p$ and
$g \left(x\right) = 2 {x}^{2} + 5 x + 3$

we need $k , p$ such that for suitable $b , c$

$f \left(x\right) = g \left(x\right) \left(b x + c\right)$

Equating coefficients we have the conditions

{ (-3 c + p =0), ( 7 - 3 b - 5 c = 0), (5 + 2 c + k = 0), (2+2b=0) :}

Solving we have

$\left\{b = - 1 , c = 2 , k = 1 , p = 6\right\}$

so

$\left\{k = 1 , p = 6\right\}$