Find the remainder when $x^{3} - 4 x^{2} + 12 x + 7$ is divided by $x + \frac{1}{2}$ .

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Solution

Given, $f (x) = x^{3} - 4 x^{2} + 12 x + 7$ .

Here, $f (x)$ is divided by $(x + \frac{1}{2})$ .

Now, according to the remainder theorem, when $f (x)$ is divided by $(x + \frac{1}{2})$ , then the remainder is $f (- \frac{1}{2})$ .
Then,

$f (- \frac{1}{2})$ $= (- \frac{1}{2})^{3} - 4 (- \frac{1}{2})^{2} + 12 (- \frac{1}{2}) + 7$

$= (- \frac{1}{8}) - 4 (\frac{1}{4}) + 12 (- \frac{1}{2}) + 7$

$= (- \frac{1}{8}) - 1 - 6 + 7$

$= (- \frac{1}{8}) - 7 + 7$

$= - \frac{1}{8}$ .

Hence, the remainder obtained is $- \frac{1}{8}$ .

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Find the remainder when x3−4x2+12x+7 is divided by x+12.

Find the remainder when $x^{3} - 4 x^{2} + 12 x + 7$ is divided by $x + \frac{1}{2}$ .