$x + 7 = \frac{x + 1}{4 x ^{4}}$

Answer

$x_{1} = 0 x_{4} = - 1.7701 - 0.9921 i x_{2} = 1.8746 x_{5} = 0.3328 + 1.87531 i x_{3} = - 1.7701 + 0.9921 i x_{6} = 0.3328 - 1.87531 i$

Explanation

x + 7 4 x^{4} x + 4 x^{4} \cdot 7 4 x^{5} + 28 x^{4} 4 x^{5} + 28 x^{4} - x^{9} - x^{8} - x^{9} - x^{8} + 4 x^{5} + 28 x^{4} = \frac{x + 1}{4 x ^{4}} = 4 x^{4} \frac{x + 1}{4 x ^{4}} = x^{9} + x^{8} = 0 = 0 multiply ALL terms by 4 x^{4} . cancel out the denominators move all terms to the left hand side simplify left side

In order to solve $- x^{9} - x^{8} + 4 x^{5} + 28 x^{4} = 0$ , first we need to factor our $x^{4}$ .

- x^{9} - x^{8} + 4 x^{5} + 28 x^{4} = x^{4} (- x^{5} - x^{4} + 4 x + 28)

$x = 0$ is a root of multiplicity $4$ .

The remaining roots can be found by solving equation $- x^{5} - x^{4} + 4 x + 28 = 0$ .

This polynomial has no rational roots that can be found using Rational Root Test.

Roots were found using Newton method.

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x+7=4x4x+1​

x1​=0x4​=−1.7701−0.9921i​x2​=1.8746x5​=0.3328+1.87531i​x3​=−1.7701+0.9921ix6​=0.3328−1.87531i​

$x + 7 = \frac{x + 1}{4 x ^{4}}$

$x_{1} = 0 x_{4} = - 1.7701 - 0.9921 i x_{2} = 1.8746 x_{5} = 0.3328 + 1.87531 i x_{3} = - 1.7701 + 0.9921 i x_{6} = 0.3328 - 1.87531 i$