Question

Divide the polynomial f(x) = 5x2 – 8x3 + 2 – 15x by the polynomial 2x – 4x2 + 1 and verify the division algorithm.

Answer 1

Answer:. -1/3

Step-by-step explanation:

Answer 2

$Given \:f(x) = -8x^{3}+5x^{2}-15x+2 \:and \\g(x) = -4x^{2}+2x+1$

Quotient : 2x - (1/4) ___

-4x²+2x+1)-8x³+5x²-15x+2(

************* -8x³+4x²+2x

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*************** x²-17x+2

*************** x²-(x/2)-(1/4)

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***********[-33/2]x+(9/4)

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$Dividend \:f(x) = -8x^{3}+5x^{2}-15x+2$

$Divisor\:g(x) = -4x^{2}+2x+1$

$Quotient\:q(x) = 2x - \frac{1}{4}$

$Remainder\: r(x) = \frac{-33}{2}x + \frac{9}{4}$

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

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

Verification :

$g(x) \times q(x) + r(x) \\= ( -4x^{2}+2x+1)\Big(2x - \frac{1}{4}\Big) +\Big( \frac{-33}{2}x + \frac{9}{4}\Big)$

$= -4x^{2}\Big(2x - \frac{1}{4}\Big)+2x\Big(2x - \frac{1}{4}\Big)+1\Big(2x - \frac{1}{4}\Big)+\Big( \frac{-33}{2}x + \frac{9}{4}\Big)$

$=-8x^{3} + x^{2} + 4x^{2} - \frac{x}{2} +2x - \frac{1}{4} - \frac{33}{2}x + \frac{9}{4}$

$= -8x^{3} + 5x^{2} - \Big( \frac{1}{2} -2+\frac{33}{2}\Big)x + \frac{-1}{4}+ \frac{9}{4}$

$= -8x^{3} + 5x^{2} - \Big( \frac{1-4+33}{2} \Big)x + \frac{-1+9}{4}$

$= -8x^{3} + 5x^{2} - \Big( \frac{30}{2} \Big)x + \frac{8}{4}$

$= -8x^{3} + 5x^{2} - 15x + 2\\= f(x)\\= Dividend$

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