Find k such that f(x) = x^4 + kx^3 + 2 has the factor of x + 1

a.) -3
b.) -2
c.) 3
d.) 2

option c is correct k=3

first divide this expression simply according to our textbook rule

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x+1 ) x^4 + kx^3 +2 ( x^3 + (k-1)x^2 - (k-1)x + (k-1)
____x^4 + x^3

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0 + (k-1)x^3 + 2
(k-1)x^3 + (k-1)x^2

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0 - (k-1)x^2 + 2
-(k-1)x^2 - (k-1)x

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0 + (k-1)x + 2
+ (k-1)x + (k-1)
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-(k-1)+2


on dividing f(x) by x+1 x^3 + (k-1)x^2 - (k-1)x + (k-1) is the quotient and -(k-1) + 2 is remeinder

but x+1 is the factor of f(x) therefore the remeinder must be 0

remeinder = 0
-(k-1)+2 = 0
-k+1+2 = 0
-k + 3 = 0
-k = -3
k = 3

therefore option c is correct