If x-1 - x-5 then find1- x 2-1 - x 22- x4-1 - x- 2

We know (a+b)^2= a^2+2ab+b^2 given that x+1/x=5 squaring both LHS amp; RHS gives (x+1/x)^2=5^2 using the identity; (x^2)+(2*x*1/x)+(1/x)^2=25 (x^2)+2+(1/x)= ...

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Byju's Answer

Standard VIII

Mathematics

Expansion of (x^2 - y^2)

If x+1 / x=5 ...

Question

If x+1/x =5 then find
1. x²+1/x²
2. x⁴+ 1/x^2

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Solution

We know (a+b)^2= a^2+2ab+b^2
given that x+1/x=5
squaring both LHS & RHS gives (x+1/x)^2=5^2
using the identity;
(x^2)+(2x1/x)+(1/x)^2=25
(x^2)+2+(1/x)=25-2
(x^2)+(1/x)^2=23 ■

now squaring the above result (x^2)^2+(2(x^2)(1/x^2))+(1/x)^2=23^2
(x^4)+2+(1/x)^4=529
(x^4)+(1/x)^4=527■

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Q. If

x - \frac{1}{x} = 5

, find the value of

x^{2} + \frac{1}{x^{2}}

and

x^{4} + \frac{1}{x^{4}}

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x^{2} + x + 1 = 0

, then the numerical value of

{(x + \frac{1}{x})}^{2} + {(x^{2} + \frac{1}{x^{2}})}^{2} + {(x^{3} + \frac{1}{x^{3}})}^{2} + {(x^{4} + \frac{1}{x^{4}})}^{2} + . . . . + {(x^{27} + \frac{1}{x^{27}})}^{2}

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Q. The remainder when the polynomial

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is divided by

1 + x + x^{2} + x^{3} + \dots . + x^{11}

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is divided by

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is:

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x^{2} + x + 1 = 0

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If x+1/x =5 then find 1. x​​​​​​2+1/x​​​​​​2 2. x​​​​​​4 + 1/x^2

We know (a+b)^2= a^2+2ab+b^2 given that x+1/x=5 squaring both LHS & RHS gives (x+1/x)^2=5^2 using the identity; (x^2)+(2*x*1/x)+(1/x)^2=25 (x^2)+2+(1/x)=25-2 (x^2)+(1/x)^2=23 ■ now squaring the above result (x^2)^2+(2*(x^2)*(1/x^2))+(1/x)^2=23^2 (x^4)+2+(1/x)^4=529 (x^4)+(1/x)^4=527■

If x+1/x =5 then find
1. x²+1/x²
2. x⁴+ 1/x^2

We know (a+b)^2= a^2+2ab+b^2
given that x+1/x=5
squaring both LHS & RHS gives (x+1/x)^2=5^2
using the identity;
(x^2)+(2x1/x)+(1/x)^2=25
(x^2)+2+(1/x)=25-2
(x^2)+(1/x)^2=23 ■

now squaring the above result (x^2)^2+(2(x^2)(1/x^2))+(1/x)^2=23^2
(x^4)+2+(1/x)^4=529
(x^4)+(1/x)^4=527■