x^5 = 18x^3 - 81x

Understand the Problem

The question is asking to solve the polynomial equation x^5 = 18x^3 - 81x for the variable x. The solution involves rearranging the equation to set it to zero and then factoring or using algebraic methods to find the roots.

Answer

The solutions are $x = 0$, $x = 3$, and $x = -3$.

Steps to Solve

Rearranging the equation

Start by moving all terms to one side of the equation to set it to zero: $$ x^5 - 18x^3 + 81x = 0 $$

Factoring out common terms

Notice that each term has a common factor of $x$. Factor this out: $$ x(x^4 - 18x^2 + 81) = 0 $$

Setting the factored equation to zero

This gives you two cases to consider: $$ x = 0 $$ or $$ x^4 - 18x^2 + 81 = 0 $$

Substituting for easier factoring

Let $y = x^2$. Then rewrite the equation: $$ y^2 - 18y + 81 = 0 $$

Applying the quadratic formula

Use the quadratic formula, where $a = 1$, $b = -18$, and $c = 81$: $$ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

This simplifies to: $$ y = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot 81}}{2 \cdot 1} $$

Calculate the discriminant: $$ y = \frac{18 \pm \sqrt{324 - 324}}{2} $$

Solving for y

Since the discriminant is zero: $$ y = \frac{18}{2} = 9 $$

Returning to x from y

Recall that $y = x^2$, so: $$ x^2 = 9 $$

Finding the values of x

Taking the square root of both sides gives: $$ x = 3 \quad \text{or} \quad x = -3 $$

The solutions to the equation $x^5 = 18x^3 - 81x$ are $x = 0$, $x = 3$, and $x = -3$.

More Information

This polynomial equation shows that there are three roots, including a root of zero, which is quite common in polynomial equations. The other two roots are symmetrical around zero, which often occurs in equations that can be factored nicely.

Tips

A common mistake is forgetting to set the equation to zero initially or neglecting to factor out the common terms. Always ensure that every term is accounted for when transforming the equation.

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