Question

Divide the polynomial $p\left(x\right)$ by the polynomial $g\left(x\right)$ and find the quotient and remainder in each of the following:
(i) $p\left(x\right)=x^3-3x^2+5x-3,g\left(x\right)=x^2-2$
(ii) $p\left(x\right)=x^4-3x^2+4x+5,g\left(x\right)=x^2+1-x$
(iii) $p\left(x\right)=x^4-5x+6,g\left(x\right)=2-x^2$

Answer 1

(i) $x-3$

$x^2-2)\overline{x^3-3x^2+5x-3}$

$x^3$ $-2x$

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$-3x^2+7x-3$

$-3x^2$ $-6$

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$7x-9$

Quotient$=x-3$

Remainder$=7x-9$

(ii) $x^2+x-3$

$x^2-x+1)\overline{x^4-3x^2+4x+5+0x^3}$

$x^4+x^2$ $-x^3$

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$x^3+4x^2+4x+5$

$x^3-x^2+x$

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$-3x^2+3x+5$

$-3x^2+3x-3$

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$8$

Quotient$=x^2+x-3$

Remainder$=8$

(iii) $-x^2-2$

$2-x^2)\overline{x^4+0x^3+0x^2-5x+6}$

$x^4$ $-2x^2$

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$0x^3+2x^2-5x+6$

$+2x^2$ $-4$

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$-5x+10$

Quotient$=-x^2-2$

Remainder$=-5x+10$.