Find the remainder when x^{3}+3x^{2}+3x+1 is | vedclass

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(B) According to the Remainder Theorem,if a polynomial $p(x)$ is divided by $(x - a)$,the remainder is $p(a)$.Here,$p(x) = x^{3} + 3x^{2} + 3x + 1$ an

Vedclass · Accepted Answer

$\frac{27}{8}$

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(B) According to the Remainder Theorem,if a polynomial $p(x)$ is divided by $(x - a)$,the remainder is $p(a)$.Here,$p(x) = x^{3} + 3x^{2} + 3x + 1$ and the divisor is $x - \frac{1}{2}$.Setting the divisor to zero,we get $x - \frac{1}{2} = 0$,which implies $x = \frac{1}{2}$.To find the remainder,we calculate $p\left(\frac{1}{2}\right)$:$p\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^{3} + 3\left(\frac{1}{2}\right)^{2} + 3\left(\frac{1}{2}\right) + 1$$= \frac{1}{8} + 3\left(\frac{1}{4}\right) + \frac{3}{2} + 1$$= \frac{1}{8} + \frac{3}{4} + \frac{3}{2} + 1$Taking the least common multiple $(LCM)$ as $8$:$= \frac{1 + 6 + 12 + 8}{8} = \frac{27}{8}$.Thus,the required remainder is $\frac{27}{8}$.

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

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