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

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Answer 1 · 2016-07-13T20:55:54

${k = 1, p = 6}$ ${k = 1, p = 6}$

Calling

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

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

$f (x) = g (x) (b x + c)$ $f(x)=g(x)(b x+c)$

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

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

so

${k = 1, p = 6}$

Find the values of k and p so that the expression -2x^3-kx^2+7x+p−2x3−kx2+7x+p-2x^3-kx^2+7x+p can be divided by 2x^2+5x+32x2+5x+32x^2+5x+3 without leavng any remainder.?