Find the Roots/Zeros Using the Rational Roots Test 6X^3-47x^2+36x-7=0

$6 X^{3} - 47 x^{2} + 36 x - 7 = 0$

Step 1

Move all terms not containing $X$ to the right side of the equation.

Add $47 x^{2}$ to both sides of the equation.

$6 X^{3} + 36 x - 7 = 47 x^{2}$

Subtract $36 x$ from both sides of the equation.

$6 X^{3} - 7 = 47 x^{2} - 36 x$

Add $7$ to both sides of the equation.

$6 X^{3} = 47 x^{2} - 36 x + 7$

Step 2

Divide each term in $6 X^{3} = 47 x^{2} - 36 x + 7$ by $6$ and simplify.

Divide each term in $6 X^{3} = 47 x^{2} - 36 x + 7$ by $6$ .

$\frac{6 X^{3}}{6} = \frac{47 x^{2}}{6} + \frac{- 36 x}{6} + \frac{7}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 X^{3}}{6} = \frac{47 x^{2}}{6} + \frac{- 36 x}{6} + \frac{7}{6}$

Divide $X^{3}$ by $1$ .

$X^{3} = \frac{47 x^{2}}{6} + \frac{- 36 x}{6} + \frac{7}{6}$

Simplify the right side.

Cancel the common factor of $- 36$ and $6$ .

Factor $6$ out of $- 36 x$ .

$X^{3} = \frac{47 x^{2}}{6} + \frac{6 (- 6 x)}{6} + \frac{7}{6}$

Cancel the common factors.

Factor $6$ out of $6$ .

$X^{3} = \frac{47 x^{2}}{6} + \frac{6 (- 6 x)}{6 (1)} + \frac{7}{6}$

Cancel the common factor.

$X^{3} = \frac{47 x^{2}}{6} + \frac{6 (- 6 x)}{6 \cdot 1} + \frac{7}{6}$

Rewrite the expression.

$X^{3} = \frac{47 x^{2}}{6} + \frac{- 6 x}{1} + \frac{7}{6}$

Divide $- 6 x$ by $1$ .

$X^{3} = \frac{47 x^{2}}{6} - 6 x + \frac{7}{6}$

Step 3

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

$X = \sqrt[3]{\frac{47 x^{2}}{6} - 6 x + \frac{7}{6}}$

Step 4

Simplify $\sqrt[3]{\frac{47 x^{2}}{6} - 6 x + \frac{7}{6}}$ .

To write $- 6 x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$X = \sqrt[3]{\frac{47 x^{2}}{6} - 6 x \cdot \frac{6}{6} + \frac{7}{6}}$

Combine $- 6 x$ and $\frac{6}{6}$ .

$X = \sqrt[3]{\frac{47 x^{2}}{6} + \frac{- 6 x \cdot 6}{6} + \frac{7}{6}}$

Combine the numerators over the common denominator.

$X = \sqrt[3]{\frac{47 x^{2} - 6 x \cdot 6}{6} + \frac{7}{6}}$

Combine the numerators over the common denominator.

$X = \sqrt[3]{\frac{47 x^{2} - 6 x \cdot 6 + 7}{6}}$

Multiply $6$ by $- 6$ .

$X = \sqrt[3]{\frac{47 x^{2} - 36 x + 7}{6}}$

Rewrite $\sqrt[3]{\frac{47 x^{2} - 36 x + 7}{6}}$ as $\frac{\sqrt[3]{47 x^{2} - 36 x + 7}}{\sqrt[3]{6}}$ .

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7}}{\sqrt[3]{6}}$

Multiply $\frac{\sqrt[3]{47 x^{2} - 36 x + 7}}{\sqrt[3]{6}}$ by $\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$ .

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7}}{\sqrt[3]{6}} \cdot \frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{47 x^{2} - 36 x + 7}}{\sqrt[3]{6}}$ by $\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$ .

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} {\sqrt[3]{6}}^{2}}{\sqrt[3]{6} {\sqrt[3]{6}}^{2}}$

Raise $\sqrt[3]{6}$ to the power of $1$ .

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} {\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{1} {\sqrt[3]{6}}^{2}}$

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

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} {\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{1 + 2}}$

Add $1$ and $2$ .

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} {\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{3}}$

Rewrite ${\sqrt[3]{6}}^{3}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} {\sqrt[3]{6}}^{2}}{{(6^{\frac{1}{3}})}^{3}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} {\sqrt[3]{6}}^{2}}{6^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} {\sqrt[3]{6}}^{2}}{6^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} {\sqrt[3]{6}}^{2}}{6^{\frac{3}{3}}}$

Rewrite the expression.

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} {\sqrt[3]{6}}^{2}}{6^{1}}$

Evaluate the exponent.

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} {\sqrt[3]{6}}^{2}}{6}$

Simplify the numerator.

Rewrite ${\sqrt[3]{6}}^{2}$ as ${(6^{2})}^{\frac{1}{3}}$ .

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} \sqrt[3]{6^{2}}}{6}$

Raise $6$ to the power of $2$ .

$X = \frac{\sqrt[3]{47 x^{2} - 36 x + 7} \sqrt[3]{36}}{6}$

Simplify with factoring out.

Combine using the product rule for radicals.

$X = \frac{\sqrt[3]{(47 x^{2} - 36 x + 7) \cdot 36}}{6}$

Reorder factors in $\frac{\sqrt[3]{(47 x^{2} - 36 x + 7) \cdot 36}}{6}$ .

$X = \frac{\sqrt[3]{36 (47 x^{2} - 36 x + 7)}}{6}$

Step 5

Find the Roots/Zeros Using the Rational Roots Test 6X^3-47x^2+36x-7=0

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