Precalculus Examples

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

Step 1

Write $x^{4} - 2 x^{3} + x^{2}$ as an equation.

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

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 $x$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

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

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

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

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

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

Step 2.2.2.1.2

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

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

Step 2.2.2.1.3

Multiply by $1$ .

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

Step 2.2.2.1.4

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

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

Step 2.2.2.1.5

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

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

Step 2.2.2.2

Factor using the perfect square rule.

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

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

$2 x = 2 \cdot x \cdot 1$

Step 2.2.2.2.3

Rewrite the polynomial.

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

Step 2.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

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 $0$ .

$x^{2} = 0$

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

Step 2.2.4

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

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Step 2.2.4.1

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

$x^{2} = 0$

Step 2.2.4.2

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

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Step 2.2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.4.2.2.1

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

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

Step 2.2.4.2.2.2

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

$x = \pm 0$

Step 2.2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.2.5

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set ${(x - 1)}^{2}$ equal to $0$ .

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

Step 2.2.5.2

Solve ${(x - 1)}^{2} = 0$ for $x$ .

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Step 2.2.5.2.1

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

$x - 1 = 0$

Step 2.2.5.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.2.6

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

$x = 0, 1$

Step 2.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

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

Step 3.2.2

Remove parentheses.

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

Step 3.2.3

Remove parentheses.

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

Step 3.2.4

Remove parentheses.

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

Step 3.2.5

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

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Step 3.2.5.1

Simplify each term.

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Step 3.2.5.1.1

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

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

Step 3.2.5.1.2

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

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

Step 3.2.5.1.3

Multiply $- 2$ by $0$ .

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

Step 3.2.5.1.4

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

$y = 0 + 0 + 0$

Step 3.2.5.2

Simplify by adding numbers.

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Step 3.2.5.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 3.2.5.2.2

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): $(0, 0), (1, 0)$

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

Step 5