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Let P(x) be a cubic polynomial such that P(0) = -3 and P(1) = 4. When P(x) is divided by x^2 + x + 1, the remainder is 2x + 3. What is the quotient when P(x) is divided by x^2 + x + 1?

 Jun 11, 2021
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Since $P(x)$ is cubic and $x^2+x+1$ is quadratic, the quotient must be linear.   Let the quotient be $ax+b$ for some constants $a$ and $b$.
Then
$$P(x) = (x^2+x+1)(ax+b) + 2x+3.$$
We are given that $P(0)= -3$, so
$$(0^2+0+1)(a(0)+b) + 2(0)+3 = -3$$
that is,
$$b+3=-3,$$
giving $b=-6$.
We are also given that $P(1) = 4$, so
$$(1^2+1+1)(a(1)-6) + 2(1)+3 = 4$$
that is,
$$3(a-6)+5=4$$
giving $a=\frac{17}{3}$.
Therefore, the quotient is $\frac{17}{3}x-6$.

 Jun 11, 2021

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