Question

factorize x^11+x^10+x^9...+x+1

Answer 1

$\huge\green{\mathfrak{\underline{\underline{ANSWER}}}}$

${x}^{11} + {x}^{10} + {x}^{9} + {x}^{8} + {x}^{7} + {x}^{6} + {x}^{5} + {x}^{4} + {x}^{3} + {x}^{2} + x + 1$

Let x = - 1

${ (- 1)}^{11} + {( - 1)}^{10} + {( - 1)}^{9} + { ( - 1)}^{8} + {( - 1)}^{7} + {( - 1)}^{6} + {( - 1)}^{5} + {( - 1)}^{4} + {( - 1)}^{3} + {( - 1)}^{2} + ( - 1) + 1 \\ = > 0$

Hence, x - 1 is the factor.

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Answer 2

Answer:

(x+1)(x^2+1)(x^4-x^2+1)(x^2-x+1)(x^2+x+1)

Step-by-step explanation:

x^11+x^10+x^9...+x+1

There are 12 terms. Showing in pairs as below and taking factor (x+1):

(x^11+x^10)+(x^9+x^8)+(x^7+x^6)+(x^5+x^4)+(x^3+x^2)+(x+1)=

=(x+1)(x^10+x^8+x^6+x^4+x^2+1)= (x+1)((x^10+x^8)+(x^6+x^4)+(x^2+1))

Repeating same for the second factor:

(x+1)(x^2+1)(x^8+x^4+1)

Going further:

(x+1)(x^2+1)(x^8+2x^4+1 -x^4)=

=(x+1)(x^2+1)((x^4+1)^2-x^4)=(x+1)(x^2+1)(x^4+x^2+1)(x^4-x^2+1)

Once more:

(x+1)(x^2+1)(x^4-x^2+1)(x^4+2x^2+1-x^2)=

=(x+1)(x^2+1)(x^4-x^2+1)((x^2+1)^2 -x^2))=

=(x+1)(x^2+1)(x^4-x^2+1)(x^2-x+1)(x^2+x+1)

This is the final one, we got 5 factors as above

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