Identify the Zeros and Their Multiplicities f(x)=2x^3-4x^2-34x+68

$f (x) = 2 x^{3} - 4 x^{2} - 34 x + 68$

To find the roots/zeros, set $2 x^{3} - 4 x^{2} - 34 x + 68$ equal to $0$ and solve.

$2 x^{3} - 4 x^{2} - 34 x + 68 = 0$

Factor the left side of the equation.

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Factor $2$ out of $2 x^{3} - 4 x^{2} - 34 x + 68$ .

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Factor $2$ out of $2 x^{3}$ .

$2 (x^{3}) - 4 x^{2} - 34 x + 68 = 0$

Factor $2$ out of $- 4 x^{2}$ .

$2 (x^{3}) + 2 (- 2 x^{2}) - 34 x + 68 = 0$

Factor $2$ out of $- 34 x$ .

$2 (x^{3}) + 2 (- 2 x^{2}) + 2 (- 17 x) + 68 = 0$

Factor $2$ out of $68$ .

$2 x^{3} + 2 (- 2 x^{2}) + 2 (- 17 x) + 2 \cdot 34 = 0$

Factor $2$ out of $2 x^{3} + 2 (- 2 x^{2})$ .

$2 (x^{3} - 2 x^{2}) + 2 (- 17 x) + 2 \cdot 34 = 0$

Factor $2$ out of $2 (x^{3} - 2 x^{2}) + 2 (- 17 x)$ .

$2 (x^{3} - 2 x^{2} - 17 x) + 2 \cdot 34 = 0$

Factor $2$ out of $2 (x^{3} - 2 x^{2} - 17 x) + 2 \cdot 34$ .

$2 (x^{3} - 2 x^{2} - 17 x + 34) = 0$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$2 ((x^{3} - 2 x^{2}) - 17 x + 34) = 0$

Factor out the greatest common factor (GCF) from each group.

$2 (x^{2} (x - 2) - 17 (x - 2)) = 0$

Factor.

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Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$2 ((x - 2) (x^{2} - 17)) = 0$

Remove unnecessary parentheses.

$2 (x - 2) (x^{2} - 17) = 0$

Divide each term by $2$ and simplify.

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Divide each term in $2 (x - 2) (x^{2} - 17) = 0$ by $2$ .

$\frac{2 (x - 2) (x^{2} - 17)}{2} = \frac{0}{2}$

Simplify $\frac{2 (x - 2) (x^{2} - 17)}{2}$ .

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 (x - 2) (x^{2} - 17)}{2} = \frac{0}{2}$

Divide $(x - 2) (x^{2} - 17)$ by $1$ .

$(x - 2) (x^{2} - 17) = \frac{0}{2}$

Expand $(x - 2) (x^{2} - 17)$ using the FOIL Method.

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Apply the distributive property.

$x (x^{2} - 17) - 2 (x^{2} - 17) = \frac{0}{2}$

Apply the distributive property.

$x \cdot x^{2} + x \cdot - 17 - 2 (x^{2} - 17) = \frac{0}{2}$

Apply the distributive property.

$x \cdot x^{2} + x \cdot - 17 - 2 x^{2} - 2 \cdot - 17 = \frac{0}{2}$

Simplify each term.

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Multiply $x$ by $x^{2}$ by adding the exponents.

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Multiply $x$ by $x^{2}$ .

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Raise $x$ to the power of $1$ .

$x^{1} x^{2} + x \cdot - 17 - 2 x^{2} - 2 \cdot - 17 = \frac{0}{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + 2} + x \cdot - 17 - 2 x^{2} - 2 \cdot - 17 = \frac{0}{2}$

Add $1$ and $2$ .

$x^{3} + x \cdot - 17 - 2 x^{2} - 2 \cdot - 17 = \frac{0}{2}$

Move $- 17$ to the left of $x$ .

$x^{3} - 17 \cdot x - 2 x^{2} - 2 \cdot - 17 = \frac{0}{2}$

Multiply $- 2$ by $- 17$ .

$x^{3} - 17 x - 2 x^{2} + 34 = \frac{0}{2}$

Divide $0$ by $2$ .

$x^{3} - 17 x - 2 x^{2} + 34 = 0$

Factor the left side of the equation.

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Reorder terms.

$x^{3} - 2 x^{2} - 17 x + 34 = 0$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$(x^{3} - 2 x^{2}) - 17 x + 34 = 0$

Factor out the greatest common factor (GCF) from each group.

$x^{2} (x - 2) - 17 (x - 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$(x - 2) (x^{2} - 17) = 0$

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

$x - 2 = 0$

$x^{2} - 17 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$x^{2} - 17 = 0$

Add $17$ to both sides of the equation.

$x^{2} = 17$

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

$x = \pm \sqrt{17}$

The complete solution is the result of both the positive and negative portions of the solution.

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First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{17}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{17}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{17}, - \sqrt{17}$

The final solution is all the values that make $(x - 2) (x^{2} - 17) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 2$ (Multiplicity of $1$ )

$x = \sqrt{17}$ (Multiplicity of $1$ )

$x = - \sqrt{17}$ (Multiplicity of $1$ )

Identify the Zeros and Their Multiplicities f(x)=2x^3-4x^2-34x+68

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