Question

please answer the question please​

Answer 1

We know a polynomial p(x) leaves remainder p(a) on division by (x - a).

Here we have p(x) = x³ + (k + 8)x + k.

p(x) leaves remainders p(2) and p(-1) each on division by (x - 2) and (x + 1) respectively.

Given that p(2) + p(-1) = 0.

Or p(2) = - p(-1).

So, first let's find both p(2) and p(-1).

p(2) = 2³ + (k + 8)2 + k

p(2) = 8 + 2k + 16 + k

p(2) = 3k + 24

And,

p(-1) = (- 1)³ - (k + 8) + k

p(-1) = - 1 - k - 8 + k

p(-1) = - 9

Now,

p(2) = - p(-1)

3k + 24 = 9

k = - 5