Question

`x^7+x^5 = 81` ?

Answer 1

This septic equation has exactly one real root:

`x ~~ 1.80302`

Explanation:

Given:

`x^7+x^5 = 81`

Subtract `81` from both sides to get into standard form:

`x^7+x^5-81 = 0`

By the rational roots theorem, any rational roots of this septic equation are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `81` and `q` a divisor of the coefficient `1` of the leading term.

That means that the only possible rational roots are:

`+-1, +-3, +-9, +-27, +-81`

Note that the signs of the coefficients are in the pattern `+ + -`. With one change of sign, Descartes' Rule of Signs tells us that this equation has exactly `1` positive real root.

Reversing the signs on the terms of odd degree, we get the pattern `- - -`. With no changes of sign, Descartes' Rule of Signs tells us that this equation has no negative real zeros.

We find:

`(color(blue)(1))^7+(color(blue)(1))^5-81 = 1+1-81 = -79 < 0`

`(color(blue)(3))^7+(color(blue)(3))^5-81 = 2187+243-81 = 2349 > 0`

So the equation has no rational roots. It has exactly one positive irrational real root in `(1, 3)` and three complex conjugate pairs of non-real roots.

In common with most polynomials of degree `5` or more, the roots are not expresible in terms of `n`th roots (i.e. square root, cube root, etc.).

About the best we can do is find numerical approximations such as:

`x ~~ 1.80302`

graph{x^7+x^5-81 [-5, 5, -1000, 1000]}

To find approximations to the real root, you would probably use Newton's method. To get all `7` roots the Durand-Kerner method would be more useful as it finds all `7` roots at once.

For Newton's method, define:

`f(x) = x^7+x^5-81`

Then:

`f'(x) = 7x^6+5x^4`

Given an approximation `a_i` to a zero of `f(x)`, a better approximation is given by:

`a_(i+1) = a_i - (f(a_i))/(f'(a_i)) = a_i - (a_i^7+a_i^5-81)/(7a_i^6+5a_i^4)`

Putting this formula into a spreadsheet and setting `a_0 = 2`, we find:

`a_0 = 2`

`a_1 ~~ 1.85037878787879`

`a_2 ~~ 1.80631325284238`

`a_3 ~~ 1.80303898643730`

`a_4 ~~ 1.80302212305041`

`a_5 ~~ 1.80302212260564`

`a_6 ~~ 1.80302212260564`