Is (x - 1) a factor of x^20-x^19+3x^18-2x^17-x^16+3x^15+x^14+2x^10-x^9+x^6-8x^4+x^3-1

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To find if (x - 1) is a factor of x^20-x^19+3x^18-2x^17-x^16+3x^15+x^14+2x^10-x^9+x^6-8x^4+x^3-1, use the remainder theorem. According to the theorem if (x - a) is a factor of f(x), f(a) = 0.

Substitute x = 1 in x^20-x^19+3x^18-2x^17-x^16+3x^15+x^14+2x^10-x^9+x^6-8x^4+x^3-1

=> (1)^20-(1)^19+3(1)^18-2(1)^17-(1)^16+3(1)^15+(1)^14+2(1)^10-(1)^9+(1)^6-8(1)^4+(1)^3-1

=> 1 - 1 + 3 - 2 - 1 +...

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To find if (x - 1) is a factor of x^20-x^19+3x^18-2x^17-x^16+3x^15+x^14+2x^10-x^9+x^6-8x^4+x^3-1, use the remainder theorem. According to the theorem if (x - a) is a factor of f(x), f(a) = 0.

Substitute x = 1 in x^20-x^19+3x^18-2x^17-x^16+3x^15+x^14+2x^10-x^9+x^6-8x^4+x^3-1

=> (1)^20-(1)^19+3(1)^18-2(1)^17-(1)^16+3(1)^15+(1)^14+2(1)^10-(1)^9+(1)^6-8(1)^4+(1)^3-1

=> 1 - 1 + 3 - 2 - 1 + 3 + 1 + 2 - 1 + 1 - 8 + 1 - 1

=> -2

Therefore (x - 1) is not a factor of the given expression.

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