Algebra Examples

$f (x) = x {(4 x + 1)}^{2} (x - 6) (x - 6) (x^{2} + 1) (x^{2} - 6)$

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 $x$ .

$0 = x {(4 x + 1)}^{2} (x - 6) (x - 6) (x^{2} + 1) (x^{2} - 6)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x {(4 x + 1)}^{2} (x - 6) (x - 6) (x^{2} + 1) (x^{2} - 6) = 0$ .

$x {(4 x + 1)}^{2} (x - 6) (x - 6) (x^{2} + 1) (x^{2} - 6) = 0$

Step 1.2.2

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

$x = 0$

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

$x - 6 = 0$

$x^{2} + 1 = 0$

$x^{2} - 6 = 0$

Step 1.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.4

Set ${(4 x + 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set ${(4 x + 1)}^{2}$ equal to $0$ .

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

Step 1.2.4.2

Solve ${(4 x + 1)}^{2} = 0$ for $x$ .

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Step 1.2.4.2.1

Set the $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Step 1.2.4.2.2

Solve for $x$ .

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Step 1.2.4.2.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 1.2.4.2.2.2

Divide each term in $4 x = - 1$ by $4$ and simplify.

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Step 1.2.4.2.2.2.1

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 1.2.4.2.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.4.2.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 1.2.4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Step 1.2.4.2.2.2.3

Simplify the right side.

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Step 1.2.4.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Step 1.2.5

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 1.2.5.2

Add $6$ to both sides of the equation.

$x = 6$

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

Set $x^{2} - 6$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $x^{2} - 6$ equal to $0$ .

$x^{2} - 6 = 0$

Step 1.2.7.2

Solve $x^{2} - 6 = 0$ for $x$ .

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Step 1.2.7.2.1

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 1.2.7.2.2

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

$x = \pm \sqrt{6}$

Step 1.2.7.2.3

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

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Step 1.2.7.2.3.1

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

$x = \sqrt{6}$

Step 1.2.7.2.3.2

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

$x = - \sqrt{6}$

Step 1.2.7.2.3.3

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

$x = \sqrt{6}, - \sqrt{6}$

Step 1.2.8

The final solution is all the values that make $x {(4 x + 1)}^{2} (x - 6) (x - 6) (x^{2} + 1) (x^{2} - 6) = 0$ true.

$x = 0, - \frac{1}{4}, 6, i, - i, \sqrt{6}, - \sqrt{6}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- \frac{1}{4}, 0), (6, 0), (\sqrt{6}, 0), (- \sqrt{6}, 0)$

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 $y$ .

$y = (0) {(4 (0) + 1)}^{2} ((0) - 6) ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2

Solve the equation.

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Step 2.2.1

Multiply $0$ by ${(4 (0) + 1)}^{2}$ .

$y = 0 {(4 (0) + 1)}^{2} ((0) - 6) ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.2

Remove parentheses.

$y = 0 {(4 (0) + 1)}^{2} (0 - 6) ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.3

Remove parentheses.

$y = 0 {(4 (0) + 1)}^{2} (0 - 6) (0 - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.4

Remove parentheses.

$y = 0 {(4 (0) + 1)}^{2} (0 - 6) (0 - 6) (0^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.5

Remove parentheses.

$y = 0 {(4 (0) + 1)}^{2} (0 - 6) (0 - 6) (0^{2} + 1) (0^{2} - 6)$

Step 2.2.6

Remove parentheses.

$y = (0) {(4 (0) + 1)}^{2} ((0) - 6) ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.7

Simplify $(0) {(4 (0) + 1)}^{2} ((0) - 6) ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$ .

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Step 2.2.7.1

Multiply $4$ by $0$ .

$y = 0 {(0 + 1)}^{2} ((0) - 6) ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.7.2

Add $0$ and $1$ .

$y = 0 \cdot 1^{2} ((0) - 6) ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.7.3

One to any power is one.

$y = 0 \cdot 1 ((0) - 6) ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.7.4

Multiply $0$ by $1$ .

$y = 0 ((0) - 6) ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.7.5

Subtract $6$ from $0$ .

$y = 0 \cdot - 6 ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.7.6

Multiply $0$ by $- 6$ .

$y = 0 ((0) - 6) ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.7.7

Subtract $6$ from $0$ .

$y = 0 \cdot - 6 ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.7.8

Multiply $0$ by $- 6$ .

$y = 0 ({(0)}^{2} + 1) ({(0)}^{2} - 6)$

Step 2.2.7.9

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

$y = 0 (0 + 1) ({(0)}^{2} - 6)$

Step 2.2.7.10

Add $0$ and $1$ .

$y = 0 \cdot 1 ({(0)}^{2} - 6)$

Step 2.2.7.11

Multiply $0$ by $1$ .

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

Step 2.2.7.12

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

$y = 0 (0 - 6)$

Step 2.2.7.13

Subtract $6$ from $0$ .

$y = 0 \cdot - 6$

Step 2.2.7.14

Multiply $0$ by $- 6$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- \frac{1}{4}, 0), (6, 0), (\sqrt{6}, 0), (- \sqrt{6}, 0)$

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

Step 4