Algebra Examples

$4 x^{3} - 4 x - x^{5}$

Step 1

Write $4 x^{3} - 4 x - x^{5}$ as an equation.

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

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 4 x^{3} - 4 x - x^{5}$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $4 x^{3} - 4 x - x^{5} = 0$ .

$4 x^{3} - 4 x - x^{5} = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $- x$ out of $4 x^{3} - 4 x - x^{5}$ .

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Step 2.2.2.1.1

Reorder the expression.

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Step 2.2.2.1.1.1

Move $- 4 x$ .

$4 x^{3} - x^{5} - 4 x = 0$

Step 2.2.2.1.1.2

Reorder $4 x^{3}$ and $- x^{5}$ .

$- x^{5} + 4 x^{3} - 4 x = 0$

Step 2.2.2.1.2

Factor $- x$ out of $- x^{5}$ .

$- x \cdot x^{4} + 4 x^{3} - 4 x = 0$

Step 2.2.2.1.3

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

$- x \cdot x^{4} - x (- 4 x^{2}) - 4 x = 0$

Step 2.2.2.1.4

Factor $- x$ out of $- 4 x$ .

$- x \cdot x^{4} - x (- 4 x^{2}) - x \cdot 4 = 0$

Step 2.2.2.1.5

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

$- x (x^{4} - 4 x^{2}) - x \cdot 4 = 0$

Step 2.2.2.1.6

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

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

Step 2.2.2.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 2.2.2.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- x (u^{2} - 4 u + 4) = 0$

Step 2.2.2.4

Factor using the perfect square rule.

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Step 2.2.2.4.1

Rewrite $4$ as $2^{2}$ .

$- x (u^{2} - 4 u + 2^{2}) = 0$

Step 2.2.2.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot u \cdot 2$

Step 2.2.2.4.3

Rewrite the polynomial.

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

Step 2.2.2.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 2$ .

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

Step 2.2.2.5

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 2.2.3

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

$x = 0$

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

Step 2.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.5

Set ${(x^{2} - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set ${(x^{2} - 2)}^{2}$ equal to $0$ .

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

Step 2.2.5.2

Solve ${(x^{2} - 2)}^{2} = 0$ for $x$ .

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Step 2.2.5.2.1

Set the $x^{2} - 2$ equal to $0$ .

$x^{2} - 2 = 0$

Step 2.2.5.2.2

Solve for $x$ .

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Step 2.2.5.2.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 2.2.5.2.2.2

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

$x = \pm \sqrt{2}$

Step 2.2.5.2.2.3

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

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Step 2.2.5.2.2.3.1

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

$x = \sqrt{2}$

Step 2.2.5.2.2.3.2

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

$x = - \sqrt{2}$

Step 2.2.5.2.2.3.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 2.2.6

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

$x = 0, \sqrt{2}, - \sqrt{2}$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\sqrt{2}, 0), (- \sqrt{2}, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 4 {(0)}^{3} - 4 \cdot 0 - {(0)}^{5}$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = 4 \cdot 0^{3} - 4 \cdot 0 - {(0)}^{5}$

Step 3.2.2

Remove parentheses.

$y = 4 \cdot 0^{3} - 4 \cdot 0 - 0^{5}$

Step 3.2.3

Remove parentheses.

$y = 4 {(0)}^{3} - 4 \cdot 0 - {(0)}^{5}$

Step 3.2.4

Simplify $4 {(0)}^{3} - 4 \cdot 0 - {(0)}^{5}$ .

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

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

$y = 4 \cdot 0 - 4 \cdot 0 - {(0)}^{5}$

Step 3.2.4.1.2

Multiply $4$ by $0$ .

$y = 0 - 4 \cdot 0 - {(0)}^{5}$

Step 3.2.4.1.3

Multiply $- 4$ by $0$ .

$y = 0 + 0 - {(0)}^{5}$

Step 3.2.4.1.4

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

$y = 0 + 0 - 0$

Step 3.2.4.1.5

Multiply $- 1$ by $0$ .

$y = 0 + 0 + 0$

Step 3.2.4.2

Simplify by adding numbers.

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Step 3.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 3.2.4.2.2

Add $0$ and $0$ .

$y = 0$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 4

List the intersections.

x-intercept(s): $(0, 0), (\sqrt{2}, 0), (- \sqrt{2}, 0)$

y-intercept(s): $(0, 0)$

Step 5