Question

if x+1/x = 4 then find (x^4+1/x^2)/x^2-3x+1

Answer 1

Given, x + 1/x = 4

we have to find the value of $\large{\frac{\left(x^4+\frac{1}{x^2}\right)}{x^2-3x+1}}$

now, x + 1/x = 4

x² + 1 = 4x

x² - 3x + 1 = -3x + 4x = x

hence, x² - 3x + 1 = x ..........(1)

again, x⁴ + 1/x²
= (x^6 + 1)/x²
= {(x²)³ + 1³}/x²
we know, (a³ + b³) = (a + b)³ - 3ab(a + b)
so, (x²)³ + 1³ = (x² + 1)³ - 3x²(x² + 1)

now, x + 1/x = 4
x² + 1 = 4x , put it above .

(x²)³ + 1³ = (4x)³ - 3x²(4x) = 64x³ - 12x³ = 52x³

so, $\frac{\left(x^4+\frac{1}{x^2}\right)}{x^2-3x+1}$

= {52x³/x²}/x .
= 52

hence, answer is 52

Answer 2

Heya Mate

$> x + \frac{1}{x} = 4$

$= > {x}^{2} - 3x + 1 = x$

$= > \frac{ {x}^{4} + \frac{1}{ {x}^{2} } }{ {x}^{2} - 3x + 1 } = \frac{ {x}^{4} + \frac{1}{ {x}^{2} } }{x} = {x}^{3} + \frac{1}{ {x}^{3} }$

$= (x + \frac{1}{x} ) {}^{3} - 3(x + \frac{1}{x} ) = 52$

Hope that wasn't much Simplification ^^"

Have a great day