Identify the Zeros and Their Multiplicities x^9-81x^7=0

$x^{9} - 81 x^{7} = 0$

Step 1

Factor the left side of the equation.

Factor $x^{7}$ out of $x^{9} - 81 x^{7}$ .

Factor $x^{7}$ out of $x^{9}$ .

$x^{7} x^{2} - 81 x^{7} = 0$

Factor $x^{7}$ out of $- 81 x^{7}$ .

$x^{7} x^{2} + x^{7} \cdot - 81 = 0$

Factor $x^{7}$ out of $x^{7} x^{2} + x^{7} \cdot - 81$ .

$x^{7} (x^{2} - 81) = 0$

Rewrite $81$ as $9^{2}$ .

$x^{7} (x^{2} - 9^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 9$ .

$x^{7} ((x + 9) (x - 9)) = 0$

Remove unnecessary parentheses.

$x^{7} (x + 9) (x - 9) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{7} = 0$

$x + 9 = 0$

$x - 9 = 0$

Step 3

Set $x^{7}$ equal to $0$ and solve for $x$ .

Set $x^{7}$ equal to $0$ .

$x^{7} = 0$

Solve $x^{7} = 0$ for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[7]{0}$

Simplify $\sqrt[7]{0}$ .

Rewrite $0$ as $0^{7}$ .

$x = \sqrt[7]{0^{7}}$

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 4

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 5

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Step 6

The final solution is all the values that make $x^{7} (x + 9) (x - 9) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 0$ (Multiplicity of $7$ )

$x = - 9$ (Multiplicity of $1$ )

$x = 9$ (Multiplicity of $1$ )

Step 7

Identify the Zeros and Their Multiplicities x^9-81x^7=0

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