Polynomial Equation: Solving for x with 3 Solutions | Math Homework

[matematikuvol]

Homework Statement

Find $$x$$
$$(17-x)^2(11-x)+256-32(17-x)-64(11-x)=0$$

Homework Equations

The Attempt at a Solution

This eq has 3 solutions. I solved this by multiplication. Is this some other easier way. Perhaps to group some of the factors $$17-x$$ and $$11-x$$. Tnx for the answer.

 

[SammyS]

Homework Statement

Find $$x$$
$$(17-x)^2(11-x)+256-32(17-x)-64(11-x)=0$$

Homework Equations

The Attempt at a Solution

This eq has 3 solutions. I solved this by multiplication. Is this some other easier way. Perhaps to group some of the factors $$17-x$$ and $$11-x$$. Tnx for the answer.

I don't know of any particularly easy way. When multiplied out you have some pretty big coefficients and a very large constant term.

Use some substitution. Either of the following is suggested by the problem itself.

$u=17-x$

$v=11-x$​

The first takes a little less computation. It results in:

$u^3-6u^2-96u+640=0$​

The other results in a much smaller constant term.

$v^3+12v^2-60v+64=0$​

Of course if you use the substitution, $t=9-x\,,$ you get a very simple polynomial.