Algebra Examples

$f (x) = 2 x^{4} + 3 x^{3} + 8 x^{2} + 12 x$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = 2 x^{4} + 3 x^{3} + 8 x^{2} + 12 x$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x^{4} + 3 x^{3} + 8 x^{2} + 12 x = 0$ .

$2 x^{4} + 3 x^{3} + 8 x^{2} + 12 x = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $x$ out of $2 x^{4} + 3 x^{3} + 8 x^{2} + 12 x$ .

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Step 1.2.2.1.1

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

$x (2 x^{3}) + 3 x^{3} + 8 x^{2} + 12 x = 0$

Step 1.2.2.1.2

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

$x (2 x^{3}) + x (3 x^{2}) + 8 x^{2} + 12 x = 0$

Step 1.2.2.1.3

Factor $x$ out of $8 x^{2}$ .

$x (2 x^{3}) + x (3 x^{2}) + x (8 x) + 12 x = 0$

Step 1.2.2.1.4

Factor $x$ out of $12 x$ .

$x (2 x^{3}) + x (3 x^{2}) + x (8 x) + x \cdot 12 = 0$

Step 1.2.2.1.5

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

$x (2 x^{3} + 3 x^{2}) + x (8 x) + x \cdot 12 = 0$

Step 1.2.2.1.6

Factor $x$ out of $x (2 x^{3} + 3 x^{2}) + x (8 x)$ .

$x (2 x^{3} + 3 x^{2} + 8 x) + x \cdot 12 = 0$

Step 1.2.2.1.7

Factor $x$ out of $x (2 x^{3} + 3 x^{2} + 8 x) + x \cdot 12$ .

$x (2 x^{3} + 3 x^{2} + 8 x + 12) = 0$

Step 1.2.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.2.1

Group the first two terms and the last two terms.

$x ((2 x^{3} + 3 x^{2}) + 8 x + 12) = 0$

Step 1.2.2.2.2

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

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

Step 1.2.2.3

Factor.

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Step 1.2.2.3.1

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

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

Step 1.2.2.3.2

Remove unnecessary parentheses.

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

Step 1.2.3

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

$x = 0$

$2 x + 3 = 0$

$x^{2} + 4 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $2 x + 3$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 1.2.5.2

Solve $2 x + 3 = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 1.2.5.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 1.2.6

Set $x^{2} + 4$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Step 1.2.6.2

Solve $x^{2} + 4 = 0$ for $x$ .

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Step 1.2.6.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 1.2.6.2.2

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

$x = \pm \sqrt{- 4}$

Step 1.2.6.2.3

Simplify $\pm \sqrt{- 4}$ .

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Step 1.2.6.2.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 1.2.6.2.3.2

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

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 1.2.6.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{4}$

Step 1.2.6.2.3.4

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

$x = \pm i \cdot \sqrt{2^{2}}$

Step 1.2.6.2.3.5

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

$x = \pm i \cdot 2$

Step 1.2.6.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 1.2.6.2.4

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

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Step 1.2.6.2.4.1

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

$x = 2 i$

Step 1.2.6.2.4.2

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

$x = - 2 i$

Step 1.2.6.2.4.3

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

$x = 2 i, - 2 i$

Step 1.2.7

The final solution is all the values that make $x (2 x + 3) (x^{2} + 4) = 0$ true.

$x = 0, - \frac{3}{2}, 2 i, - 2 i$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- \frac{3}{2}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = 2 {(0)}^{4} + 3 {(0)}^{3} + 8 {(0)}^{2} + 12 (0)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \cdot 0^{4} + 3 {(0)}^{3} + 8 {(0)}^{2} + 12 (0)$

Step 2.2.2

Remove parentheses.

$y = 2 \cdot 0^{4} + 3 \cdot 0^{3} + 8 {(0)}^{2} + 12 (0)$

Step 2.2.3

Remove parentheses.

$y = 2 \cdot 0^{4} + 3 \cdot 0^{3} + 8 \cdot 0^{2} + 12 (0)$

Step 2.2.4

Remove parentheses.

$y = 2 {(0)}^{4} + 3 {(0)}^{3} + 8 {(0)}^{2} + 12 (0)$

Step 2.2.5

Simplify $2 {(0)}^{4} + 3 {(0)}^{3} + 8 {(0)}^{2} + 12 (0)$ .

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Step 2.2.5.1

Simplify each term.

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Step 2.2.5.1.1

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

$y = 2 \cdot 0 + 3 {(0)}^{3} + 8 {(0)}^{2} + 12 (0)$

Step 2.2.5.1.2

Multiply $2$ by $0$ .

$y = 0 + 3 {(0)}^{3} + 8 {(0)}^{2} + 12 (0)$

Step 2.2.5.1.3

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

$y = 0 + 3 \cdot 0 + 8 {(0)}^{2} + 12 (0)$

Step 2.2.5.1.4

Multiply $3$ by $0$ .

$y = 0 + 0 + 8 {(0)}^{2} + 12 (0)$

Step 2.2.5.1.5

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

$y = 0 + 0 + 8 \cdot 0 + 12 (0)$

Step 2.2.5.1.6

Multiply $8$ by $0$ .

$y = 0 + 0 + 0 + 12 (0)$

Step 2.2.5.1.7

Multiply $12$ by $0$ .

$y = 0 + 0 + 0 + 0$

Step 2.2.5.2

Simplify by adding numbers.

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Step 2.2.5.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 0$

Step 2.2.5.2.2

Add $0$ and $0$ .

$y = 0 + 0$

Step 2.2.5.2.3

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- \frac{3}{2}, 0)$

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

Step 4