Question
Find the remainder when $x^3 + 3x^2 + 3x + 1$ is divided by $5 + 2x.$
✓
Answer
Let $p(x) = x^3 + 3x^2 + 3x + 1$
$5 + 2x = 0$
$\Rightarrow2x = -5$
$\Rightarrow x = -\frac{5}{2}$
$= (-\frac{5}{2})^3 + 3(-\frac{5}{2})^2 + 3(-\frac{5}{2}) + 1$
$= -\frac{125}{8} + \frac{75}{4} - \frac{15}{2} + 1$
$= -\frac{27}{8}$
$5 + 2x = 0$
$\Rightarrow2x = -5$
$\Rightarrow x = -\frac{5}{2}$
$= (-\frac{5}{2})^3 + 3(-\frac{5}{2})^2 + 3(-\frac{5}{2}) + 1$
$= -\frac{125}{8} + \frac{75}{4} - \frac{15}{2} + 1$
$= -\frac{27}{8}$
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