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Question Number 213123 by a.lgnaoui last updated on 30/Oct/24

factoriser x^5 +x^3 +x^2 −2x−1

$$\mathrm{factoriser} \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{5}} +\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{\mathrm{x}}−\mathrm{1} \\ $$

Answered by MrGaster last updated on 30/Oct/24

(x^5 +x^3 +x^2 −2x−1)=(x−1)(x^4 +x^3 +2x^2 +3x+1) (x^4 +x^3 +2x^2 +3x+1)=(x+1)(x^3 +2x+1) (x^3 +2x+1)=(x+1)(x^2 −x+1) (x^5 +x^3 +x^2 −2x−1)=(x−1)(x+1)^2 (x^2 −x+1)

$$\left({x}^{\mathrm{5}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{1}\right)=\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{4}} +{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{1}\right) \\ $$$$\left({x}^{\mathrm{4}} +{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{1}\right)=\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{3}} +\mathrm{2}{x}+\mathrm{1}\right) \\ $$$$\left({x}^{\mathrm{3}} +\mathrm{2}{x}+\mathrm{1}\right)=\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right) \\ $$$$\left({x}^{\mathrm{5}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{1}\right)=\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right) \\ $$

Commented by a.lgnaoui last updated on 30/Oct/24

thank you.

$$\mathrm{thank}\:\mathrm{you}. \\ $$

Commented by A5T last updated on 30/Oct/24

=x^5 −x+x^3 −x+x^2 −1 =x(x^2 −1)(x^2 +1)+x(x^2 −1)+x^2 −1 =(x−1)(x+1)(x^3 +2x+1) x^3 +2x+1≠(x+1)(x^2 −x+1)

$$={x}^{\mathrm{5}} −{x}+{x}^{\mathrm{3}} −{x}+{x}^{\mathrm{2}} −\mathrm{1} \\ $$$$={x}\left({x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{1}\right)+{x}\left({x}^{\mathrm{2}} −\mathrm{1}\right)+{x}^{\mathrm{2}} −\mathrm{1} \\ $$$$=\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{3}} +\mathrm{2}{x}+\mathrm{1}\right) \\ $$$${x}^{\mathrm{3}} +\mathrm{2}{x}+\mathrm{1}\neq\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right) \\ $$

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