Question

please refer to attachment​

Answer 1

$\Huge\boxed{\textrm{(3) 0}}$

$\Large\textrm{Solution}$

$\textbf{- Remainder Theorem}$

"The substitution of the solution of a linear polynomial obtains the remainder of the division."

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Given equation is,

$\rm a^{1/3}+b^{1/3}+c^{1/3}=0$

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$\rm\therefore c^{1/3}=-(a^{1/3}+b^{1/3})$

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By cubing,

$\boxed{\rm(A+B)^{3}=A^{3}+3A^{2}B+3AB^{2}+B^{3}}$

$\rm\therefore c=-(a+3a^{2/3}b^{1/3}+3a^{1/3}b^{2/3}+b)$

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By remainder theorem,

$\rm a+b+c+3a^{1/3}b^{2/3}+3a^{2/3}b^{1/3}$

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$\rm=(a+3a^{1/3}b^{2/3}+3a^{2/3}b^{1/3}+b)-(a+3a^{2/3}b^{1/3}+3a^{1/3}b^{2/3}+b)$

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$\rm=\boxed{0}$

Hence, option (3) is correct.

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$\Large\textrm{Learn More}$

$\textbf{- Factor Theorem}$

$\boxed{\rm{\rm f(\alpha)=0\iff f(x)=(x-\alpha)Q(x)}}$

Let's interpret this as words. Getting no remainder in division means the dividend is divisible by the divisor.

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The remainder theorem gives no remainder, so it is divisible.

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$\textbf{- The Use of Factor Theorem}$

For example,

$\rm f(x)=x^{3}+7x-8$

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$\rm f(1)=0$

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$\rm\therefore f(x)=(x-1)Q(x)$

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This is how the factor theorem applies in the division of polynomials to find the factor.

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From then on, we can apply synthetic division to find another factor.

$\rm\therefore f(x)=(x-1)(x^{2}+x+8)$