Maths Question

By remainder theorem,
since polynomial P(x) is divided by x+1 , remainder is -7.
so P(-1) = -7
since polynomial P(x) is divided by x+2, remainder is -25.
so P(-2) = -25

Let P(x) = (x+1)(x+2) Q(x) + R(x) , where Q(x) is the quotient and R(x) is the remainder.

since P(x) is divided by (x+1)(x+2), deg [(x+1)(x+2)] = 2 , so deg [R(x)] < 2,
Hence, let R(x) = ax + b,

since P(-1) = (-1+1)(-1+2) Q(x) + a(-1) + b = -a + b = -7
and P(-2) = (-2+1)(-2+2) Q(x) + a(-2) + b = -2a + b = -25

so, a = 18, b = 11,
R(x) = 18x + 11,
i.e. the remainder is 18x + 11.

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