Question

Divide 3x^3 - 8x^2+3x +2 by x^2-3x +2 and verify the division algorithm

Answer 1

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Pre-Algebra

Divide (3x^3-5x^2-4x-8)/(2x^2+x)

Factor out of .

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Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .

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Divide the highest order term in the dividend by the highest order term in divisor .

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Multiply the new quotient term by the divisor.

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The expression needs to be subtracted from the dividend, so change all the signs in

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After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

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Pull the next terms from the original dividend down into the current dividend.

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Divide the highest order term in the dividend by the highest order term in divisor .

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Multiply the new quotient term by the divisor.

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The expression needs to be subtracted from the dividend, so change all the signs in

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After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

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The final answer is the quotient plus the remainder over the divisor.

Answer 2

Concept

In the division algorithm, when the number "a" is divided by the number "b", the quotient is "q", the remainder is "r", and a = bq + r. Where 0 ≤ r  This is also known as the "Euclidean Division Lemma". The division algorithm can be easily expressed as:

 Dividend = divisor x quotient + remainder

Given

We have given a polynomial $3x^3 - 8x^2+3x +2$ which divided by another polynomial $x^2-3x +2$ .

Find

We are asked to prove the division algorithm of the given polynomial.

Solution

On dividing $3x^3 - 8x^2+3x +2$ by $x^2-3x +2$ we get $3x+1$ as quotient and no remainder.

According to division algorithm

$Dividend=Divisor\times Quotient +Remainder$ ....(1)

Putting $Dividend=3x^3 - 8x^2+3x +2 , \ Divisor = x^2-3x +2 ,\ Quotient = 3x+1 , \ Remainder=0$

in equation (1) , we get

$3x^3 - 8x^2+3x +2= (x^2-3x +2)(3x+1)+0$

We need to prove LHS = RHS

$RHS= (x^2-3x +2)(3x+1)+0\\=x^2(3x+1)-3x(3x+1)+2(3x+1)\\=3x^3+x^2-9x^2-3x+6x+2\\=3x^3-8x^2+3x+2$

LHS = RHS

Hence , proved the division algorithm of the given polynoimal.

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