Calculus Examples

$2 y^{3} + y^{2} - y^{5} = x^{4} - 2 x^{3} + x^{2}$

Step 1

Set each solution of $2 y^{3} + y^{2} - y^{5}$ as a function of $x$ .

$y = x^{4} - 2 x^{3} + x^{2} \to f (x) = x^{4} - 2 x^{3} + x^{2}$

Step 2

Because the $y$ variable in the equation $2 y^{3} + y^{2} - y^{5} = x^{4} - 2 x^{3} + x^{2}$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 2.1

Differentiate both sides of the equation.

$\frac{d}{d x} (2 y^{3} + y^{2} - y^{5}) = \frac{d}{d x} (x^{4} - 2 x^{3} + x^{2})$

Step 2.2

Differentiate the left side of the equation.

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Step 2.2.1

By the Sum Rule, the derivative of $2 y^{3} + y^{2} - y^{5}$ with respect to $x$ is $\frac{d}{d x} [2 y^{3}] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- y^{5}]$ .

$\frac{d}{d x} [2 y^{3}] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- y^{5}]$

Step 2.2.2

Evaluate $\frac{d}{d x} [2 y^{3}]$ .

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Step 2.2.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 y^{3}$ with respect to $x$ is $2 \frac{d}{d x} [y^{3}]$ .

$2 \frac{d}{d x} [y^{3}] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- y^{5}]$

Step 2.2.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$2 (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [y]) + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- y^{5}]$

Step 2.2.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$2 (3 {u_{1}}^{2} \frac{d}{d x} [y]) + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- y^{5}]$

Step 2.2.2.2.3

Replace all occurrences of $u_{1}$ with $y$ .

$2 (3 y^{2} \frac{d}{d x} [y]) + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- y^{5}]$

Step 2.2.2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$2 (3 y^{2} y') + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- y^{5}]$

Step 2.2.2.4

Multiply $3$ by $2$ .

$6 y^{2} y' + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- y^{5}]$

Step 2.2.3

Evaluate $\frac{d}{d x} [y^{2}]$ .

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Step 2.2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 2.2.3.1.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$6 y^{2} y' + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y] + \frac{d}{d x} [- y^{5}]$

Step 2.2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$6 y^{2} y' + 2 u_{2} \frac{d}{d x} [y] + \frac{d}{d x} [- y^{5}]$

Step 2.2.3.1.3

Replace all occurrences of $u_{2}$ with $y$ .

$6 y^{2} y' + 2 y \frac{d}{d x} [y] + \frac{d}{d x} [- y^{5}]$

Step 2.2.3.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$6 y^{2} y' + 2 y y' + \frac{d}{d x} [- y^{5}]$

Step 2.2.4

Evaluate $\frac{d}{d x} [- y^{5}]$ .

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Step 2.2.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- y^{5}$ with respect to $x$ is $- \frac{d}{d x} [y^{5}]$ .

$6 y^{2} y' + 2 y y' - \frac{d}{d x} [y^{5}]$

Step 2.2.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = y$ .

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Step 2.2.4.2.1

To apply the Chain Rule, set $u_{3}$ as $y$ .

$6 y^{2} y' + 2 y y' - (\frac{d}{d u_{3}} [{u_{3}}^{5}] \frac{d}{d x} [y])$

Step 2.2.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 5$ .

$6 y^{2} y' + 2 y y' - (5 {u_{3}}^{4} \frac{d}{d x} [y])$

Step 2.2.4.2.3

Replace all occurrences of $u_{3}$ with $y$ .

$6 y^{2} y' + 2 y y' - (5 y^{4} \frac{d}{d x} [y])$

Step 2.2.4.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$6 y^{2} y' + 2 y y' - (5 y^{4} y')$

Step 2.2.4.4

Multiply $5$ by $- 1$ .

$6 y^{2} y' + 2 y y' - 5 y^{4} y'$

Step 2.3

Differentiate the right side of the equation.

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Step 2.3.1

Differentiate.

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Step 2.3.1.1

By the Sum Rule, the derivative of $x^{4} - 2 x^{3} + x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [x^{2}]$

Step 2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [x^{2}]$

Step 2.3.2

Evaluate $\frac{d}{d x} [- 2 x^{3}]$ .

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Step 2.3.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

$4 x^{3} - 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}]$

Step 2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 x^{3} - 2 (3 x^{2}) + \frac{d}{d x} [x^{2}]$

Step 2.3.2.3

Multiply $3$ by $- 2$ .

$4 x^{3} - 6 x^{2} + \frac{d}{d x} [x^{2}]$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$4 x^{3} - 6 x^{2} + 2 x$

Step 2.4

Reform the equation by setting the left side equal to the right side.

$6 y^{2} y' + 2 y y' - 5 y^{4} y' = 4 x^{3} - 6 x^{2} + 2 x$

Step 2.5

Solve for $y'$ .

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Step 2.5.1

Factor $y y'$ out of $6 y^{2} y' + 2 y y' - 5 y^{4} y'$ .

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Step 2.5.1.1

Factor $y y'$ out of $6 y^{2} y'$ .

$y y' (6 y) + 2 y y' - 5 y^{4} y' = 4 x^{3} - 6 x^{2} + 2 x$

Step 2.5.1.2

Factor $y y'$ out of $2 y y'$ .

$y y' (6 y) + y y' \cdot 2 - 5 y^{4} y' = 4 x^{3} - 6 x^{2} + 2 x$

Step 2.5.1.3

Factor $y y'$ out of $- 5 y^{4} y'$ .

$y y' (6 y) + y y' \cdot 2 + y y' (- 5 y^{3}) = 4 x^{3} - 6 x^{2} + 2 x$

Step 2.5.1.4

Factor $y y'$ out of $y y' (6 y) + y y' \cdot 2$ .

$y y' (6 y + 2) + y y' (- 5 y^{3}) = 4 x^{3} - 6 x^{2} + 2 x$

Step 2.5.1.5

Factor $y y'$ out of $y y' (6 y + 2) + y y' (- 5 y^{3})$ .

$y y' (6 y + 2 - 5 y^{3}) = 4 x^{3} - 6 x^{2} + 2 x$

Step 2.5.2

Reorder terms.

$y y' (- 5 y^{3} + 6 y + 2) = 4 x^{3} - 6 x^{2} + 2 x$

Step 2.5.3

Divide each term in $y y' (- 5 y^{3} + 6 y + 2) = 4 x^{3} - 6 x^{2} + 2 x$ by $y (- 5 y^{3} + 6 y + 2)$ and simplify.

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Step 2.5.3.1

Divide each term in $y y' (- 5 y^{3} + 6 y + 2) = 4 x^{3} - 6 x^{2} + 2 x$ by $y (- 5 y^{3} + 6 y + 2)$ .

$\frac{y y' (- 5 y^{3} + 6 y + 2)}{y (- 5 y^{3} + 6 y + 2)} = \frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} + \frac{- 6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)}$

Step 2.5.3.2

Simplify the left side.

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Step 2.5.3.2.1

Cancel the common factor of $y$ .

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Step 2.5.3.2.1.1

Cancel the common factor.

$\frac{y y' (- 5 y^{3} + 6 y + 2)}{y (- 5 y^{3} + 6 y + 2)} = \frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} + \frac{- 6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)}$

Step 2.5.3.2.1.2

Rewrite the expression.

$\frac{y' (- 5 y^{3} + 6 y + 2)}{- 5 y^{3} + 6 y + 2} = \frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} + \frac{- 6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)}$

Step 2.5.3.2.2

Cancel the common factor of $- 5 y^{3} + 6 y + 2$ .

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Step 2.5.3.2.2.1

Cancel the common factor.

$\frac{y' (- 5 y^{3} + 6 y + 2)}{- 5 y^{3} + 6 y + 2} = \frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} + \frac{- 6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)}$

Step 2.5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} + \frac{- 6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)}$

Step 2.5.3.3

Simplify the right side.

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Step 2.5.3.3.1

Move the negative in front of the fraction.

$y' = \frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)}$

Step 2.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)}$

Step 3

Set the derivative equal to $0$ then solve the equation $\frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} = 0$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y (- 5 y^{3} + 6 y + 2), y (- 5 y^{3} + 6 y + 2), y (- 5 y^{3} + 6 y + 2), 1$

Step 3.1.2

Factor $y (- 5 y^{3} + 6 y + 2)$ .

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Step 3.1.2.1

Factor.

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Step 3.1.2.1.1

Factor $- 1$ out of $- 5 y^{3} + 6 y + 2$ .

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Step 3.1.2.1.1.1

Factor $- 1$ out of $- 5 y^{3}$ .

$y (- (5 y^{3}) + 6 y + 2)$

Step 3.1.2.1.1.2

Factor $- 1$ out of $6 y$ .

$y (- (5 y^{3}) - (- 6 y) + 2)$

Step 3.1.2.1.1.3

Rewrite $2$ as $- 1 (- 2)$ .

$y (- (5 y^{3}) - (- 6 y) - 1 (- 2))$

Step 3.1.2.1.1.4

Factor $- 1$ out of $- (5 y^{3}) - (- 6 y)$ .

$y (- (5 y^{3} - 6 y) - 1 (- 2))$

Step 3.1.2.1.1.5

Factor $- 1$ out of $- (5 y^{3} - 6 y) - 1 (- 2)$ .

$y (- (5 y^{3} - 6 y - 2))$

Step 3.1.2.1.2

Remove unnecessary parentheses.

$y \cdot - 1 (5 y^{3} - 6 y - 2)$

Step 3.1.2.2

Factor out negative.

$- (y (5 y^{3} - 6 y - 2))$

Step 3.1.2.3

Remove unnecessary parentheses.

$- y (5 y^{3} - 6 y - 2)$

Step 3.1.3

Since $- y (5 y^{3} - 6 y - 2), - y (5 y^{3} - 6 y - 2), - y (5 y^{3} - 6 y - 2), 1$ contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $- y (5 y^{3} - 6 y - 2), - y (5 y^{3} - 6 y - 2), - y (5 y^{3} - 6 y - 2), 1$ are:

1. Find the LCM for the numeric part $- 1, - 1, - 1, 1$ .

2. Find the LCM for the variable part $y^{1}, y^{1}, y^{1}$ .

3. Find the LCM for the compound variable part $5 y^{3} - 6 y - 2, 5 y^{3} - 6 y - 2, 5 y^{3} - 6 y - 2$ .

4. Multiply each LCM together.

Step 3.1.4

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 3.1.5

Since the LCM is the smallest positive number, $LCM (- 1, - 1, - 1, 1) = LCM (1, 1, 1, 1)$

Step 3.1.6

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 3.1.7

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 3.1.8

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 3.1.9

The LCM of $y^{1}, y^{1}, y^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y$

Step 3.1.10

The factor for $5 y^{3} - 6 y - 2$ is $5 y^{3} - 6 y - 2$ itself.

$(5 y^{3} - 6 y - 2) = 5 y^{3} - 6 y - 2$

$(5 y^{3} - 6 y - 2)$ occurs $1$ time.

Step 3.1.11

The LCM of $5 y^{3} - 6 y - 2, 5 y^{3} - 6 y - 2, 5 y^{3} - 6 y - 2$ is the result of multiplying all factors the greatest number of times they occur in either term.

$5 y^{3} - 6 y - 2$

Step 3.1.12

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$y (5 y^{3} - 6 y - 2)$

Step 3.2

Multiply each term in $\frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} = 0$ by $y (5 y^{3} - 6 y - 2)$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} = 0$ by $y (5 y^{3} - 6 y - 2)$ .

$\frac{4 x^{3}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Cancel the common factor of $y$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

Step 3.2.2.1.1.2

Rewrite the expression.

$\frac{4 x^{3}}{- 5 y^{3} + 6 y + 2} (5 y^{3} - 6 y - 2) - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.2

Multiply $\frac{4 x^{3}}{- 5 y^{3} + 6 y + 2}$ by $5 y^{3} - 6 y - 2$ .

$\frac{4 x^{3} (5 y^{3} - 6 y - 2)}{- 5 y^{3} + 6 y + 2} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.3

Cancel the common factor of $5 y^{3} - 6 y - 2$ and $- 5 y^{3} + 6 y + 2$ .

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Step 3.2.2.1.3.1

Factor $- 1$ out of $5 y^{3}$ .

$\frac{4 x^{3} (- (- 5 y^{3}) - 6 y - 2)}{- 5 y^{3} + 6 y + 2} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.3.2

Factor $- 1$ out of $- 6 y$ .

$\frac{4 x^{3} (- (- 5 y^{3}) - (6 y) - 2)}{- 5 y^{3} + 6 y + 2} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.3.3

Factor $- 1$ out of $- (- 5 y^{3}) - (6 y)$ .

$\frac{4 x^{3} (- (- 5 y^{3} + 6 y) - 2)}{- 5 y^{3} + 6 y + 2} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.3.4

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{4 x^{3} (- (- 5 y^{3} + 6 y) - 1 (2))}{- 5 y^{3} + 6 y + 2} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.3.5

Factor $- 1$ out of $- (- 5 y^{3} + 6 y) - 1 (2)$ .

$\frac{4 x^{3} (- (- 5 y^{3} + 6 y + 2))}{- 5 y^{3} + 6 y + 2} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.3.6

Rewrite $- (- 5 y^{3} + 6 y + 2)$ as $- 1 (- 5 y^{3} + 6 y + 2)$ .

$\frac{4 x^{3} (- 1 (- 5 y^{3} + 6 y + 2))}{- 5 y^{3} + 6 y + 2} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.3.7

Cancel the common factor.

Step 3.2.2.1.3.8

Divide $4 x^{3} \cdot (- 1)$ by $1$ .

$4 x^{3} \cdot (- 1) - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.4

Multiply $- 1$ by $4$ .

$- 4 x^{3} - \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.5

Cancel the common factor of $y$ .

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Step 3.2.2.1.5.1

Move the leading negative in $- \frac{6 x^{2}}{y (- 5 y^{3} + 6 y + 2)}$ into the numerator.

$- 4 x^{3} + \frac{- 6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.5.2

Cancel the common factor.

$- 4 x^{3} + \frac{- 6 x^{2}}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.5.3

Rewrite the expression.

$- 4 x^{3} + \frac{- 6 x^{2}}{- 5 y^{3} + 6 y + 2} (5 y^{3} - 6 y - 2) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.6

Move the negative in front of the fraction.

$- 4 x^{3} - \frac{(6) x^{2}}{- 5 y^{3} + 6 y + 2} (5 y^{3} - 6 y - 2) + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.7

Apply the distributive property.

$- 4 x^{3} - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} (5 y^{3}) - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} (- 6 y) - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} \cdot - 2 + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8

Simplify.

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Step 3.2.2.1.8.1

Multiply $- \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} (5 y^{3})$ .

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Step 3.2.2.1.8.1.1

Multiply $5$ by $- 1$ .

$- 4 x^{3} - 5 \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} y^{3} - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} (- 6 y) - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} \cdot - 2 + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8.1.2

Combine $- 5$ and $\frac{6 x^{2}}{- 5 y^{3} + 6 y + 2}$ .

$- 4 x^{3} + \frac{- 5 (6 x^{2})}{- 5 y^{3} + 6 y + 2} y^{3} - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} (- 6 y) - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} \cdot - 2 + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8.1.3

Multiply $6$ by $- 5$ .

$- 4 x^{3} + \frac{- 30 x^{2}}{- 5 y^{3} + 6 y + 2} y^{3} - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} (- 6 y) - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} \cdot - 2 + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8.1.4

Combine $\frac{- 30 x^{2}}{- 5 y^{3} + 6 y + 2}$ and $y^{3}$ .

$- 4 x^{3} + \frac{- 30 x^{2} y^{3}}{- 5 y^{3} + 6 y + 2} - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} (- 6 y) - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} \cdot - 2 + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8.2

Multiply $- \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} (- 6 y)$ .

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Step 3.2.2.1.8.2.1

Multiply $- 6$ by $- 1$ .

$- 4 x^{3} + \frac{- 30 x^{2} y^{3}}{- 5 y^{3} + 6 y + 2} + 6 \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} y - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} \cdot - 2 + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8.2.2

Combine $6$ and $\frac{6 x^{2}}{- 5 y^{3} + 6 y + 2}$ .

$- 4 x^{3} + \frac{- 30 x^{2} y^{3}}{- 5 y^{3} + 6 y + 2} + \frac{6 (6 x^{2})}{- 5 y^{3} + 6 y + 2} y - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} \cdot - 2 + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8.2.3

Multiply $6$ by $6$ .

$- 4 x^{3} + \frac{- 30 x^{2} y^{3}}{- 5 y^{3} + 6 y + 2} + \frac{36 x^{2}}{- 5 y^{3} + 6 y + 2} y - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} \cdot - 2 + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8.2.4

Combine $\frac{36 x^{2}}{- 5 y^{3} + 6 y + 2}$ and $y$ .

$- 4 x^{3} + \frac{- 30 x^{2} y^{3}}{- 5 y^{3} + 6 y + 2} + \frac{36 x^{2} y}{- 5 y^{3} + 6 y + 2} - \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} \cdot - 2 + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8.3

Multiply $- \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} \cdot - 2$ .

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Step 3.2.2.1.8.3.1

Multiply $- 2$ by $- 1$ .

$- 4 x^{3} + \frac{- 30 x^{2} y^{3}}{- 5 y^{3} + 6 y + 2} + \frac{36 x^{2} y}{- 5 y^{3} + 6 y + 2} + 2 \frac{6 x^{2}}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8.3.2

Combine $2$ and $\frac{6 x^{2}}{- 5 y^{3} + 6 y + 2}$ .

$- 4 x^{3} + \frac{- 30 x^{2} y^{3}}{- 5 y^{3} + 6 y + 2} + \frac{36 x^{2} y}{- 5 y^{3} + 6 y + 2} + \frac{2 (6 x^{2})}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.8.3.3

Multiply $6$ by $2$ .

$- 4 x^{3} + \frac{- 30 x^{2} y^{3}}{- 5 y^{3} + 6 y + 2} + \frac{36 x^{2} y}{- 5 y^{3} + 6 y + 2} + \frac{12 x^{2}}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.9

Move the negative in front of the fraction.

$- 4 x^{3} - \frac{30 x^{2} y^{3}}{- 5 y^{3} + 6 y + 2} + \frac{36 x^{2} y}{- 5 y^{3} + 6 y + 2} + \frac{12 x^{2}}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.10

Combine the numerators over the common denominator.

$- 4 x^{3} + \frac{- 30 x^{2} y^{3} + 36 x^{2} y}{- 5 y^{3} + 6 y + 2} + \frac{12 x^{2}}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.11

Combine the numerators over the common denominator.

$- 4 x^{3} + \frac{- 30 x^{2} y^{3} + 36 x^{2} y + 12 x^{2}}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.12

Factor $6 x^{2}$ out of $- 30 x^{2} y^{3} + 36 x^{2} y + 12 x^{2}$ .

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Step 3.2.2.1.12.1

Factor $6 x^{2}$ out of $- 30 x^{2} y^{3}$ .

$- 4 x^{3} + \frac{6 x^{2} (- 5 y^{3}) + 36 x^{2} y + 12 x^{2}}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.12.2

Factor $6 x^{2}$ out of $36 x^{2} y$ .

$- 4 x^{3} + \frac{6 x^{2} (- 5 y^{3}) + 6 x^{2} (6 y) + 12 x^{2}}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.12.3

Factor $6 x^{2}$ out of $12 x^{2}$ .

$- 4 x^{3} + \frac{6 x^{2} (- 5 y^{3}) + 6 x^{2} (6 y) + 6 x^{2} (2)}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.12.4

Factor $6 x^{2}$ out of $6 x^{2} (- 5 y^{3}) + 6 x^{2} (6 y)$ .

$- 4 x^{3} + \frac{6 x^{2} (- 5 y^{3} + 6 y) + 6 x^{2} (2)}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.12.5

Factor $6 x^{2}$ out of $6 x^{2} (- 5 y^{3} + 6 y) + 6 x^{2} (2)$ .

$- 4 x^{3} + \frac{6 x^{2} (- 5 y^{3} + 6 y + 2)}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.13

Cancel the common factor of $- 5 y^{3} + 6 y + 2$ .

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Step 3.2.2.1.13.1

Cancel the common factor.

$- 4 x^{3} + \frac{6 x^{2} (- 5 y^{3} + 6 y + 2)}{- 5 y^{3} + 6 y + 2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.13.2

Divide $6 x^{2}$ by $1$ .

$- 4 x^{3} + 6 x^{2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.14

Cancel the common factor of $y$ .

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Step 3.2.2.1.14.1

Cancel the common factor.

$- 4 x^{3} + 6 x^{2} + \frac{2 x}{y (- 5 y^{3} + 6 y + 2)} (y (5 y^{3} - 6 y - 2)) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.14.2

Rewrite the expression.

$- 4 x^{3} + 6 x^{2} + \frac{2 x}{- 5 y^{3} + 6 y + 2} (5 y^{3} - 6 y - 2) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.15

Multiply $\frac{2 x}{- 5 y^{3} + 6 y + 2}$ by $5 y^{3} - 6 y - 2$ .

$- 4 x^{3} + 6 x^{2} + \frac{2 x (5 y^{3} - 6 y - 2)}{- 5 y^{3} + 6 y + 2} = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.16

Cancel the common factor of $5 y^{3} - 6 y - 2$ and $- 5 y^{3} + 6 y + 2$ .

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Step 3.2.2.1.16.1

Factor $- 1$ out of $5 y^{3}$ .

$- 4 x^{3} + 6 x^{2} + \frac{2 x (- (- 5 y^{3}) - 6 y - 2)}{- 5 y^{3} + 6 y + 2} = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.16.2

Factor $- 1$ out of $- 6 y$ .

$- 4 x^{3} + 6 x^{2} + \frac{2 x (- (- 5 y^{3}) - (6 y) - 2)}{- 5 y^{3} + 6 y + 2} = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.16.3

Factor $- 1$ out of $- (- 5 y^{3}) - (6 y)$ .

$- 4 x^{3} + 6 x^{2} + \frac{2 x (- (- 5 y^{3} + 6 y) - 2)}{- 5 y^{3} + 6 y + 2} = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.16.4

Rewrite $- 2$ as $- 1 (2)$ .

$- 4 x^{3} + 6 x^{2} + \frac{2 x (- (- 5 y^{3} + 6 y) - 1 (2))}{- 5 y^{3} + 6 y + 2} = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.16.5

Factor $- 1$ out of $- (- 5 y^{3} + 6 y) - 1 (2)$ .

$- 4 x^{3} + 6 x^{2} + \frac{2 x (- (- 5 y^{3} + 6 y + 2))}{- 5 y^{3} + 6 y + 2} = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.16.6

Rewrite $- (- 5 y^{3} + 6 y + 2)$ as $- 1 (- 5 y^{3} + 6 y + 2)$ .

$- 4 x^{3} + 6 x^{2} + \frac{2 x (- 1 (- 5 y^{3} + 6 y + 2))}{- 5 y^{3} + 6 y + 2} = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.16.7

Cancel the common factor.

$- 4 x^{3} + 6 x^{2} + \frac{2 x (- 1 (- 5 y^{3} + 6 y + 2))}{- 5 y^{3} + 6 y + 2} = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.16.8

Divide $2 x \cdot (- 1)$ by $1$ .

$- 4 x^{3} + 6 x^{2} + 2 x \cdot (- 1) = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.2.1.17

Multiply $- 1$ by $2$ .

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (y (5 y^{3} - 6 y - 2))$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Apply the distributive property.

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (y (5 y^{3}) + y (- 6 y) + y \cdot - 2)$

Step 3.2.3.2

Simplify.

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Step 3.2.3.2.1

Rewrite using the commutative property of multiplication.

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (5 y \cdot y^{3} + y (- 6 y) + y \cdot - 2)$

Step 3.2.3.2.2

Rewrite using the commutative property of multiplication.

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (5 y \cdot y^{3} - 6 y \cdot y + y \cdot - 2)$

Step 3.2.3.2.3

Move $- 2$ to the left of $y$ .

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (5 y \cdot y^{3} - 6 y \cdot y - 2 \cdot y)$

Step 3.2.3.3

Simplify each term.

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Step 3.2.3.3.1

Multiply $y$ by $y^{3}$ by adding the exponents.

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Step 3.2.3.3.1.1

Move $y^{3}$ .

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (5 (y^{3} y) - 6 y \cdot y - 2 \cdot y)$

Step 3.2.3.3.1.2

Multiply $y^{3}$ by $y$ .

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Step 3.2.3.3.1.2.1

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

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (5 (y^{3} y^{1}) - 6 y \cdot y - 2 \cdot y)$

Step 3.2.3.3.1.2.2

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

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (5 y^{3 + 1} - 6 y \cdot y - 2 \cdot y)$

Step 3.2.3.3.1.3

Add $3$ and $1$ .

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (5 y^{4} - 6 y \cdot y - 2 \cdot y)$

Step 3.2.3.3.2

Multiply $y$ by $y$ by adding the exponents.

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Step 3.2.3.3.2.1

Move $y$ .

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (5 y^{4} - 6 (y \cdot y) - 2 \cdot y)$

Step 3.2.3.3.2.2

Multiply $y$ by $y$ .

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (5 y^{4} - 6 y^{2} - 2 \cdot y)$

$- 4 x^{3} + 6 x^{2} - 2 x = 0 (5 y^{4} - 6 y^{2} - 2 y)$

Step 3.2.3.4

Multiply $0$ by $5 y^{4} - 6 y^{2} - 2 y$ .

$- 4 x^{3} + 6 x^{2} - 2 x = 0$

Step 3.3

Solve the equation.

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Step 3.3.1

Factor the left side of the equation.

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Step 3.3.1.1

Factor $- 2 x$ out of $- 4 x^{3} + 6 x^{2} - 2 x$ .

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Step 3.3.1.1.1

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

$- 2 x (2 x^{2}) + 6 x^{2} - 2 x = 0$

Step 3.3.1.1.2

Factor $- 2 x$ out of $6 x^{2}$ .

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

Step 3.3.1.1.3

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

$- 2 x (2 x^{2}) - 2 x (- 3 x) - 2 x \cdot 1 = 0$

Step 3.3.1.1.4

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

$- 2 x (2 x^{2} - 3 x) - 2 x \cdot 1 = 0$

Step 3.3.1.1.5

Factor $- 2 x$ out of $- 2 x (2 x^{2} - 3 x) - 2 x (1)$ .

$- 2 x (2 x^{2} - 3 x + 1) = 0$

Step 3.3.1.2

Factor.

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Step 3.3.1.2.1

Factor by grouping.

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Step 3.3.1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = - 3$ .

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Step 3.3.1.2.1.1.1

Factor $- 3$ out of $- 3 x$ .

$- 2 x (2 x^{2} - 3 x + 1) = 0$

Step 3.3.1.2.1.1.2

Rewrite $- 3$ as $- 1$ plus $- 2$

$- 2 x (2 x^{2} + (- 1 - 2) x + 1) = 0$

Step 3.3.1.2.1.1.3

Apply the distributive property.

$- 2 x (2 x^{2} - 1 x - 2 x + 1) = 0$

Step 3.3.1.2.1.2

Factor out the greatest common factor from each group.

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Step 3.3.1.2.1.2.1

Group the first two terms and the last two terms.

$- 2 x ((2 x^{2} - 1 x) - 2 x + 1) = 0$

Step 3.3.1.2.1.2.2

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

$- 2 x (x (2 x - 1) - (2 x - 1)) = 0$

Step 3.3.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$- 2 x ((2 x - 1) (x - 1)) = 0$

Step 3.3.1.2.2

Remove unnecessary parentheses.

$- 2 x (2 x - 1) (x - 1) = 0$

Step 3.3.2

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

$x = 0$

$2 x - 1 = 0$

$x - 1 = 0$

Step 3.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 3.3.4

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 3.3.4.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 3.3.4.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 3.3.4.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 3.3.4.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 3.3.4.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 3.3.4.2.2.2

Simplify the left side.

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Step 3.3.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.3.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 3.3.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 3.3.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.3.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.3.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.3.6

The final solution is all the values that make $- 2 x (2 x - 1) (x - 1) = 0$ true.

$x = 0, \frac{1}{2}, 1$

Step 4

Solve the function $f (x) = x^{4} - 2 x^{3} + x^{2}$ at $x = 0$ .

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Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{4} - 2 {(0)}^{3} + {(0)}^{2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - 2 {(0)}^{3} + {(0)}^{2}$

Step 4.2.1.2

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - 2 \cdot 0 + {(0)}^{2}$

Step 4.2.1.3

Multiply $- 2$ by $0$ .

$f (0) = 0 + 0 + {(0)}^{2}$

Step 4.2.1.4

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + 0 + 0$

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 4.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 4.2.3

The final answer is $0$ .

$0$

Step 5

Solve the function $f (x) = x^{4} - 2 x^{3} + x^{2}$ at $x = \frac{1}{2}$ .

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Step 5.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = {(\frac{1}{2})}^{4} - 2 {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Apply the product rule to $\frac{1}{2}$ .

$f (\frac{1}{2}) = \frac{1^{4}}{2^{4}} - 2 {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{2}$

Step 5.2.1.2

One to any power is one.

$f (\frac{1}{2}) = \frac{1}{2^{4}} - 2 {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{2}$

Step 5.2.1.3

Raise $2$ to the power of $4$ .

$f (\frac{1}{2}) = \frac{1}{16} - 2 {(\frac{1}{2})}^{3} + {(\frac{1}{2})}^{2}$

Step 5.2.1.4

Apply the product rule to $\frac{1}{2}$ .

$f (\frac{1}{2}) = \frac{1}{16} - 2 \frac{1^{3}}{2^{3}} + {(\frac{1}{2})}^{2}$

Step 5.2.1.5

One to any power is one.

$f (\frac{1}{2}) = \frac{1}{16} - 2 \frac{1}{2^{3}} + {(\frac{1}{2})}^{2}$

Step 5.2.1.6

Raise $2$ to the power of $3$ .

$f (\frac{1}{2}) = \frac{1}{16} - 2 (\frac{1}{8}) + {(\frac{1}{2})}^{2}$

Step 5.2.1.7

Cancel the common factor of $2$ .

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Step 5.2.1.7.1

Factor $2$ out of $- 2$ .

$f (\frac{1}{2}) = \frac{1}{16} + 2 (- 1) (\frac{1}{8}) + {(\frac{1}{2})}^{2}$

Step 5.2.1.7.2

Factor $2$ out of $8$ .

$f (\frac{1}{2}) = \frac{1}{16} + 2 \cdot (- 1 \frac{1}{2 \cdot 4}) + {(\frac{1}{2})}^{2}$

Step 5.2.1.7.3

Cancel the common factor.

$f (\frac{1}{2}) = \frac{1}{16} + 2 \cdot (- 1 \frac{1}{2 \cdot 4}) + {(\frac{1}{2})}^{2}$

Step 5.2.1.7.4

Rewrite the expression.

$f (\frac{1}{2}) = \frac{1}{16} - 1 (\frac{1}{4}) + {(\frac{1}{2})}^{2}$

Step 5.2.1.8

Rewrite $- 1 (\frac{1}{4})$ as $- (\frac{1}{4})$ .

$f (\frac{1}{2}) = \frac{1}{16} - (\frac{1}{4}) + {(\frac{1}{2})}^{2}$

Step 5.2.1.9

Apply the product rule to $\frac{1}{2}$ .

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} + \frac{1^{2}}{2^{2}}$

Step 5.2.1.10

One to any power is one.

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} + \frac{1}{2^{2}}$

Step 5.2.1.11

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

$f (\frac{1}{2}) = \frac{1}{16} - \frac{1}{4} + \frac{1}{4}$

Step 5.2.2

Combine fractions.

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Step 5.2.2.1

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = \frac{1}{16} + \frac{- 1 + 1}{4}$

Step 5.2.2.2

Simplify the expression.

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Step 5.2.2.2.1

Add $- 1$ and $1$ .

$f (\frac{1}{2}) = \frac{1}{16} + \frac{0}{4}$

Step 5.2.2.2.2

Divide $0$ by $4$ .

$f (\frac{1}{2}) = \frac{1}{16} + 0$

Step 5.2.2.2.3

Add $\frac{1}{16}$ and $0$ .

$f (\frac{1}{2}) = \frac{1}{16}$

Step 5.2.3

The final answer is $\frac{1}{16}$ .

$\frac{1}{16}$

Step 6

Solve the function $f (x) = x^{4} - 2 x^{3} + x^{2}$ at $x = 1$ .

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Step 6.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = {(1)}^{4} - 2 {(1)}^{3} + {(1)}^{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

One to any power is one.

$f (1) = 1 - 2 {(1)}^{3} + {(1)}^{2}$

Step 6.2.1.2

One to any power is one.

$f (1) = 1 - 2 \cdot 1 + {(1)}^{2}$

Step 6.2.1.3

Multiply $- 2$ by $1$ .

$f (1) = 1 - 2 + {(1)}^{2}$

Step 6.2.1.4

One to any power is one.

$f (1) = 1 - 2 + 1$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $2$ from $1$ .

$f (1) = - 1 + 1$

Step 6.2.2.2

Add $- 1$ and $1$ .

$f (1) = 0$

Step 6.2.3

The final answer is $0$ .

$0$

Step 7

The horizontal tangent lines are $y = 0, y = \frac{1}{16}, y = 0$

$y = 0, y = \frac{1}{16}, y = 0$

Step 8