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Factor.\newlinep3+2p2−8p−16p^3 + 2p^2 - 8p - 16

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Full solution

Q. Factor.\newlinep3+2p2−8p−16p^3 + 2p^2 - 8p - 16
  1. Group and Factor: Group terms to find common factors.\newlineGroup the first two terms and the last two terms separately.\newline(p3+2p2)−(8p+16)(p^3 + 2p^2) - (8p + 16)
  2. Factor Common Terms: Factor out the greatest common factor from each group.\newlineFrom the first group, factor out p2p^2.\newlinep2(p+2)p^2(p + 2)\newlineFrom the second group, factor out −8-8.\newline−8(p+2)-8(p + 2)\newlineNow we have:\newlinep2(p+2)−8(p+2)p^2(p + 2) - 8(p + 2)
  3. Factor Common Binomial: Factor out the common binomial factor.\newlineThe common binomial factor is (p+2)(p + 2).\newlineFactor (p+2)(p + 2) out of the expression.\newline(p+2)(p2−8)(p + 2)(p^2 - 8)
  4. Recognize Difference of Squares: Recognize that p2−8p^2 - 8 is a difference of squares.\newlinep2−8p^2 - 8 can be factored as (p+8)(p−8)(p + \sqrt{8})(p - \sqrt{8}).\newlineSince 8\sqrt{8} simplifies to 222\sqrt{2}, we can write:\newline(p+22)(p−22)(p + 2\sqrt{2})(p - 2\sqrt{2})
  5. Write Final Factored Form: Write the final factored form.\newlineCombine the factored parts to get the final answer.\newline(p+2)(p+22)(p−22)(p + 2)(p + 2\sqrt{2})(p - 2\sqrt{2})

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