Divide (2x^4-3x^3+7)/(x^2-2)

Problem

(2*x^4−3*x^3+7)/(x^2−2)

Solution

1. Set up the long division by writing the dividend 2*x^4−3*x^3+0*x^2+0*x+7 and the divisor x^2−2

2. Divide the first term of the dividend 2*x^4 by the first term of the divisor x^2 to get 2*x^2

3. Multiply 2*x^2 by the divisor x^2−2 to get 2*x^4−4*x^2 then subtract this from the dividend.

(2*x^4−3*x^3+0*x^2)−(2*x^4−4*x^2)=−3*x^3+4*x^2

4. Divide the new leading term −3*x^3 by x^2 to get −3*x

5. Multiply −3*x by the divisor x^2−2 to get −3*x^3+6*x then subtract this from the current polynomial.

(−3*x^3+4*x^2+0*x)−(−3*x^3+6*x)=4*x^2−6*x

6. Divide the new leading term 4*x^2 by x^2 to get 4

7. Multiply 4 by the divisor x^2−2 to get 4*x^2−8 then subtract this from the current polynomial to find the remainder.

(4*x^2−6*x+7)−(4*x^2−8)=−6*x+15

8. Combine the quotient and the remainder over the divisor.

Final Answer

(2*x^4−3*x^3+7)/(x^2−2)=2*x^2−3*x+4+(−6*x+15)/(x^2−2)

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