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.?

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Answer 1 · 2016-07-13T20:55:54

#{k = 1, p = 6}#

Calling

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

we need #k,p# such that for suitable #b,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}#