Question

On dividing x cube -5xsquare+6x+4 by a polynomial g(x) the quotient and the remainder were x-3 and 4 respectivly find g(x)

Answer 1

I hope it was helpful.

Answer 2

$Given \: Dividend = x^{3}-5x^{2}+6x+4 ,\\Divisor = g(x) ,\\Quotient = q(x) = x - 3 , \\Remainder = r(x) = 4$

$\blue { (By \: Division \: Algorithm ) }$

$\boxed { \pink { g(x) \times q(x) + r(x) = Dividend }}$

$\implies g(x) \times (x-3) + 4 = x^{3}-5x^{2}+6x+4$

$\implies g(x) \times (x-3) = x^{3}-5x^{2}+6x+4-4$

$\implies g(x) \times (x-3) = x^{3}-5x^{2}+6x$

$\implies g(x) = \frac{x^{3}-5x^{2}+6x}{(x-3)}$

$\implies g(x) = \frac{x(x^{2}-5x+6)}{(x-3)}$

$\implies g(x) = \frac{x(x^{2}-3x-2x+6)}{(x-3)}$

$\implies g(x) = \frac{x[x(x-3) -2(x-3)]}{(x-3)}$

$\implies g(x) = \frac{x(x-3)(x-2)}{(x-3)}$

$\implies g(x) = x(x-2) \\= x^{2} - 2x$

Therefore.,

$\red { Value \:of \:g(x) } \green {= x^{2} - 2x }$

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