Precalculus Examples

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

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Rewrite the equation as $- 3 x^{6} + 2 x^{4} + x^{2} = 0$ .

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

Factor the left side of the equation.

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

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

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

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

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

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

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

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

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

Factor $- x^{2}$ out of $- x^{2} (3 x^{4} - 2 x^{2}) - x^{2} \cdot - 1$ .

$- x^{2} (3 x^{4} - 2 x^{2} - 1) = 0$

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

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

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

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

Factor by grouping.

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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 $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = - 2$ .

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Factor $- 2$ out of $- 2 u$ .

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

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

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

Apply the distributive property.

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

Multiply $u$ by $1$ .

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

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

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

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

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

Rewrite $1$ as $1^{2}$ .

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

Factor.

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Factor.

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Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

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

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

$x^{2} = 0$

$3 x^{2} + 1 = 0$

$x + 1 = 0$

$x - 1 = 0$

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

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

$x^{2} = 0$

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

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

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Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set $3 x^{2} + 1$ equal to $0$ and solve for $x$ .

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Set $3 x^{2} + 1$ equal to $0$ .

$3 x^{2} + 1 = 0$

Solve $3 x^{2} + 1 = 0$ for $x$ .

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Subtract $1$ from both sides of the equation.

$3 x^{2} = - 1$

Divide each term in $3 x^{2} = - 1$ by $3$ and simplify.

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Divide each term in $3 x^{2} = - 1$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 1}{3}$

Simplify the left side.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 1}{3}$

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

$x^{2} = \frac{- 1}{3}$

Simplify the right side.

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Move the negative in front of the fraction.

$x^{2} = - \frac{1}{3}$

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

$x = \pm \sqrt{- \frac{1}{3}}$

Simplify $\pm \sqrt{- \frac{1}{3}}$ .

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Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

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Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1}{3}}$

Rewrite $1$ as $1^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1^{2}}{3}}$

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{3}}$

One to any power is one.

$x = \pm i \sqrt{\frac{1}{3}}$

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{3}}$

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{3}}$

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Combine and simplify the denominator.

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Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

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

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$x = \pm i \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x = \pm i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$x = \pm i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm i \frac{\sqrt{3}}{3^{1}}$

Evaluate the exponent.

$x = \pm i \frac{\sqrt{3}}{3}$

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

$x = \pm \frac{i \sqrt{3}}{3}$

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.

$x = \frac{i \sqrt{3}}{3}$

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

$x = - \frac{i \sqrt{3}}{3}$

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

$x = \frac{i \sqrt{3}}{3}, - \frac{i \sqrt{3}}{3}$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $- x^{2} (3 x^{2} + 1) (x + 1) (x - 1) = 0$ true.

$x = 0, \frac{i \sqrt{3}}{3}, - \frac{i \sqrt{3}}{3}, - 1, 1$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 1, 0), (1, 0)$

Step 2

Find the y-intercepts.

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

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

Solve the equation.

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Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

$y = - 3 \cdot 0^{6} + 2 \cdot 0^{4} + 0^{2}$

Remove parentheses.

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

Simplify $- 3 {(0)}^{6} + 2 {(0)}^{4} + {(0)}^{2}$ .

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Simplify each term.

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

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

Multiply $- 3$ by $0$ .

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

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

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

Multiply $2$ by $0$ .

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

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

$y = 0 + 0 + 0$

Simplify by adding numbers.

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Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 1, 0), (1, 0)$

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

Step 4