Question

Is x+1 a factor of x^20-1 and x^51-1

Answer 1

Step-by-step explanation:

x+1=0

x= -1

if p(-1)=0

then (x+1) will factor of p(x)

p(x)= x²⁰-1

p(-1)=(-1)²⁰-1=1-1=0

hence (x+1) is factor of x²⁰-1

again, p(x)=x⁵¹-1

p(-1)=(-1)⁵¹-1=-1-1=2

hence (x+1) is not the factor of x⁵¹-1

Answer 2

$: \implies\sf \purple{x + 1 = 0} \\ : \ \implies \sf{ x= 0 - 1} \\ : \implies\sf{x = - 1}$

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$: \implies \sf \red{ {x}^{20} - 1} \\ : \implies \sf{ { (- 1)}^{20} - 1 } \\ : \implies \sf{1 - 1} \\ : \implies \sf{0}$

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$: \implies \sf \red{ {x}^{51} - 1} \\ : \implies \sf{ { (- 1)}^{51} - 1 } \\ : \implies \sf{ - 1 - 1} \\ : \implies \sf{ - 2}$

$\bf \purple{ \therefore \: {x}^{20} - 1 \: is \: a \: factor \: of \: x + 1}$

$\bf \purple{ \therefore \: {x}^{51} - 1 \: is\: not \: a \: factor \: of \: x + 1}$