Question

If x^2+bx+b is a factor of x^3+2x^2+2x+c then find the value of b-c is

Answer 1

/ * We have to find (b+c ) not (b-c) */

$Given \: Dividend\: p(x) = x^{3} + 2x^{2} +2x+ c ,\\divisor\: g(x) = x^{2}+bx+b$

/* From the attachment above , we get */

$Quotient \: q(x)= x + (2-b)$

$Remainder \:r(x) = [(2-b)-b(2-b)]x+[c-b(2-b)]$

/* According to the problem given */

$If \: g(x) \:is \: a \: factor \: p(x) \:then \: r(x) = 0$

$(2-b)-b(2-b) = 0 \implies 2-b =b(2-b) \: --(1)$

$and \: c - b(2-b) = 0 \\\implies c - (2-b) = 0 \: [ From \: (1) ]$

$\implies c - 2 + b = 0$

$\implies b + c = 2$

Therefore.,

$\red{ Value \:of \: (b+c) } \green { = 2 }$

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