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simplify r3−2r2s−4rs2+s3/4r2s−4rs2−r3r^3-2r^2s-4rs^2+s^3 / 4r^2s-4rs^2-r^3

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Full solution

Q. simplify r3−2r2s−4rs2+s3/4r2s−4rs2−r3r^3-2r^2s-4rs^2+s^3 / 4r^2s-4rs^2-r^3
  1. Factor Numerator: Factor the numerator r3−2r2s−4rs2+s3 r^3 - 2r^2s - 4rs^2 + s^3 .\newlineGroup terms: (r3−2r2s)+(−4rs2+s3) (r^3 - 2r^2s) + (-4rs^2 + s^3) .\newlineFactor out common terms: r2(r−2s)−s2(4r−s) r^2(r - 2s) - s^2(4r - s) .\newlineNotice a sign error, correct grouping: r2(r−2s)+s2(s−4r) r^2(r - 2s) + s^2(s - 4r) .\newlineRewrite with common factor: (r−2s)(r2+s2) (r - 2s)(r^2 + s^2) .
  2. Factor Denominator: Factor the denominator 4r2s−4rs2−r3 4r^2s - 4rs^2 - r^3 .\newlineRearrange terms: −r3+4r2s−4rs2 -r^3 + 4r^2s - 4rs^2 .\newlineFactor out common terms: −r(r2−4rs+4s2) -r(r^2 - 4rs + 4s^2) .\newlineRecognize a perfect square: −r(r−2s)2 -r(r - 2s)^2 .
  3. Simplify Expression: Simplify the expression by dividing the factored numerator by the factored denominator.\newline(r−2s)(r2+s2)−r(r−2s)2\frac{(r - 2s)(r^2 + s^2)}{-r(r - 2s)^2}.\newlineCancel out common terms: (r−2s) (r - 2s) in numerator and denominator.\newlineResult: r2+s2−r(r−2s)\frac{r^2 + s^2}{-r(r - 2s)}.

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