x+1 is the factor of the polynomial P(x) = x^3 + kx^2 + x + 6 if and only if the number -1 is the root of the polynomial
P(-1) = 0,
according to the Remainder theorem.
P(-1) = (-1)^3 + k*(-1)^2 + (-1) + 6 = 0
-1 + k - 1 + 6 = 0
k + 4 = 0.
k = -4. ANSWER
Solved.
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Theorem (the remainder theorem)
1. The remainder of division the polynomial by the binomial is equal to the value of the polynomial.
2. The binomial divides the polynomial if and only if the value of is the root of the polynomial , i.e. .
3. The binomial factors the polynomial if and only if the value of is the root of the polynomial , i.e. .
