Question

Divide x³-2x²-x+9 by x²+x+2 and verify the division algorithm of polynomials.​

Answer 1

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Answer 2

Solution :

We have f(x) = x³ - 2x² - x + 9 divided by p(x) = x² + x + 2.

Polynomial Division

$\boxed{\begin{array}{l|n|r}\sf x^2+ x + 2& \sf x^3-2x^2-x+9 & \sf x-3 \\ & \sf x^3 + x^2 + 2x \\ &(-)\:\:(-)\:\:(-)\\ &\rule{100}{0.8}\\ & \sf \qquad -3x^2 - 3x + 9\\ &\sf \qquad-3x^2 - 3x - 6 \\ & \qquad \:\:(+)\:\:(+)\:\:(+)\\ & \qquad \quad \rule{70}{0.8}\\ & \sf \qquad\qquad\qquad \sf \:\:\:\:\:15\end{array}}$

∴ We get remainder is 15 by division.

$\boxed{\bf{V\:E\:R\:I\:F\:I\:C\:A\:T\:I\:O\:N\::}}}$

$\mapsto\sf{Dividend=(Divisor\times Quotient)+remainder}\\\\\mapsto\sf{x^{3} - 2x^{2} - x + 9 = (x^{2} + x + 2)\times (x-3) + 15}\\\\\mapsto\sf{x^{3} - 2x^{2} - x + 9 =x^{3} + x^{2} +2x -3x^{2} -3x - 6 + 15}\\\\\mapsto\bf{x^{3} - 2x^{2} - x + 9 =x^{3} -2x^{2} -x + 9}$

Verified .