Question

what should be added to the polynomial x3 _3x2+6x_15 so that it is completely divisible by x_3

Answer 1

I thought it helps you .
cheers

Answer 2

$Let \: p(x) = x^{3} - 3x^{2} + 6x - 15$

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/* By Remainder Theorem */

If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divisible by the linear polynomial (x-a) ,then the remainder is p(a) */

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$Let \: m \: should \: be \: added \:to \: p(x) \\so \:that \: it \:is \: completely \: divisble \:by \: (x-3)$

$Now , g(x) = x^{3} - 3x^{2} + 6x - 15 + m$

$Remainder = g(3) = 0$

/* Put x = 0 in equation (1) , we get */

$\implies 3^{3} - 3\times 3^{2} + 6 \times 3 - 15 + m = 0$

$\implies 27 - 27 + 18 - 15 + m = 0$

$\implies 3 + m = 0$

$\implies m = -3$

Therefore.,

$\green { -3 \: should \:be \: added \: to \:the }\\\green { polynomial \: so \:that \: it \:is \: completely}\\\green { divisble \:by \: (x-3) }$

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