Algebra Examples

$f (x) = x^{5} - 7 x^{4} + x^{3} - 7 x^{2}$

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 $x$ .

$0 = x^{5} - 7 x^{4} + x^{3} - 7 x^{2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{5} - 7 x^{4} + x^{3} - 7 x^{2} = 0$ .

$x^{5} - 7 x^{4} + x^{3} - 7 x^{2} = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $x^{2}$ out of $x^{5} - 7 x^{4} + x^{3} - 7 x^{2}$ .

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Step 1.2.2.1.1

Factor $x^{2}$ out of $x^{5}$ .

$x^{2} x^{3} - 7 x^{4} + x^{3} - 7 x^{2} = 0$

Step 1.2.2.1.2

Factor $x^{2}$ out of $- 7 x^{4}$ .

$x^{2} x^{3} + x^{2} (- 7 x^{2}) + x^{3} - 7 x^{2} = 0$

Step 1.2.2.1.3

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

$x^{2} x^{3} + x^{2} (- 7 x^{2}) + x^{2} x - 7 x^{2} = 0$

Step 1.2.2.1.4

Factor $x^{2}$ out of $- 7 x^{2}$ .

$x^{2} x^{3} + x^{2} (- 7 x^{2}) + x^{2} x + x^{2} \cdot - 7 = 0$

Step 1.2.2.1.5

Factor $x^{2}$ out of $x^{2} x^{3} + x^{2} (- 7 x^{2})$ .

$x^{2} (x^{3} - 7 x^{2}) + x^{2} x + x^{2} \cdot - 7 = 0$

Step 1.2.2.1.6

Factor $x^{2}$ out of $x^{2} (x^{3} - 7 x^{2}) + x^{2} x$ .

$x^{2} (x^{3} - 7 x^{2} + x) + x^{2} \cdot - 7 = 0$

Step 1.2.2.1.7

Factor $x^{2}$ out of $x^{2} (x^{3} - 7 x^{2} + x) + x^{2} \cdot - 7$ .

$x^{2} (x^{3} - 7 x^{2} + x - 7) = 0$

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.

$x^{2} ((x^{3} - 7 x^{2}) + x - 7) = 0$

Step 1.2.2.2.2

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

$x^{2} (x^{2} (x - 7) + 1 (x - 7)) = 0$

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, $x - 7$ .

$x^{2} ((x - 7) (x^{2} + 1)) = 0$

Step 1.2.2.3.2

Remove unnecessary parentheses.

$x^{2} (x - 7) (x^{2} + 1) = 0$

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 $0$ .

$x^{2} = 0$

$x - 7 = 0$

$x^{2} + 1 = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x^{2} = 0$

Step 1.2.4.2

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

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Step 1.2.4.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.4.2.2.1

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

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

Step 1.2.4.2.2.2

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

$x = \pm 0$

Step 1.2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.5

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 1.2.5.2

Add $7$ to both sides of the equation.

$x = 7$

Step 1.2.6

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

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Step 1.2.6.1

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

$x^{2} + 1 = 0$

Step 1.2.6.2

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

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Step 1.2.6.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 1.2.6.2.2

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

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

Step 1.2.6.2.3

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

$x = \pm i$

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.

$x = i$

Step 1.2.6.2.4.2

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

$x = - i$

Step 1.2.6.2.4.3

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

$x = i, - i$

Step 1.2.7

The final solution is all the values that make $x^{2} (x - 7) (x^{2} + 1) = 0$ true.

$x = 0, 7, i, - i$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (7, 0)$

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 $y$ .

$y = {(0)}^{5} - 7 {(0)}^{4} + {(0)}^{3} - 7 {(0)}^{2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{5} - 7 {(0)}^{4} + {(0)}^{3} - 7 {(0)}^{2}$

Step 2.2.2

Remove parentheses.

$y = 0^{5} - 7 \cdot 0^{4} + {(0)}^{3} - 7 {(0)}^{2}$

Step 2.2.3

Remove parentheses.

$y = 0^{5} - 7 \cdot 0^{4} + 0^{3} - 7 {(0)}^{2}$

Step 2.2.4

Remove parentheses.

$y = 0^{5} - 7 \cdot 0^{4} + 0^{3} - 7 \cdot 0^{2}$

Step 2.2.5

Remove parentheses.

$y = {(0)}^{5} - 7 {(0)}^{4} + {(0)}^{3} - 7 {(0)}^{2}$

Step 2.2.6

Simplify ${(0)}^{5} - 7 {(0)}^{4} + {(0)}^{3} - 7 {(0)}^{2}$ .

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Step 2.2.6.1

Simplify each term.

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Step 2.2.6.1.1

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

$y = 0 - 7 {(0)}^{4} + {(0)}^{3} - 7 {(0)}^{2}$

Step 2.2.6.1.2

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

$y = 0 - 7 \cdot 0 + {(0)}^{3} - 7 {(0)}^{2}$

Step 2.2.6.1.3

Multiply $- 7$ by $0$ .

$y = 0 + 0 + {(0)}^{3} - 7 {(0)}^{2}$

Step 2.2.6.1.4

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

$y = 0 + 0 + 0 - 7 {(0)}^{2}$

Step 2.2.6.1.5

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

$y = 0 + 0 + 0 - 7 \cdot 0$

Step 2.2.6.1.6

Multiply $- 7$ by $0$ .

$y = 0 + 0 + 0 + 0$

Step 2.2.6.2

Simplify by adding numbers.

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Step 2.2.6.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 0$

Step 2.2.6.2.2

Add $0$ and $0$ .

$y = 0 + 0$

Step 2.2.6.2.3

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): $(0, 0), (7, 0)$

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

Step 4