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Important Algebra Formulas — Examples

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Examples

1. When $x + 1 = k$

Formula: x²+1/ x²= k²−2

Ex: If $x + 1 = 5$ , find x²+1/ x²

$Solution:$ 5²−2 = 25−2 = 23

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Formula: x³+1/ x³= k³−3k

Ex: If $x + 1 = 4$ , find x³+1/ x³

$Solution:$ 4³−3(4) = 64−12 = 52

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Formula: x⁴+1/ x⁴= k⁴− 4k²+2

Ex: If $x + 1/ x = 3$ , find x⁴+1/ x⁴.

$Solution:$ 3⁴− 4(3²) + 2 = 81 − 36 + 2 = 47

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Formula: x⁵+1/ x⁵= k⁵−5k³+5k

Example: If $x + 1 = 2$ , find x⁵+1/ x⁵.

$Solution:$ 2⁵−5(2³)+5(2) = 32−40+10 = 2

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Formula: x⁶+1/ x⁶ = k⁶−6k⁴+9k²−2

Example: If $x + 1 = 3$ , find x⁶+1/ x⁶.

$Solution:$ 3⁶− 6(3⁴) + 9(3²) − 2 729 − 486 + 81 − 2 = 322

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2. When

x - 1/ x = k

Formula: x²+1/ x²= k²+2

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Example: If $x - 1 = 4$ , find x²+1/ x²

$Solution:$ 4² $+ 2 = 16 + 2 = 18$

Formula: x³−1/ x³= k³+3k

Example: If $x - 1/ x = 3$ , find x³−1/ x³

$Solution:$ 3³+ 3(3) = 27+9 = 36

Formula: x⁴+1/ x⁴= k⁴+4k²+2

Example: If $x - 1/ x = 2$ , find x⁴+1/ x⁴

$Solution:$ 2⁴ $+ 4 ($ 2² $) + 2$ $= 16 + 16 + 2 = 34$

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