Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

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 $0$ .

$6 x + 1 = 0$

$x^{2} + 1 = 0$

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

Step 1.2.3

Set $6 x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $6 x + 1$ equal to $0$ .

$6 x + 1 = 0$

Step 1.2.3.2

Solve $6 x + 1 = 0$ for $x$ .

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Step 1.2.3.2.1

Subtract $1$ from both sides of the equation.

$6 x = - 1$

Step 1.2.3.2.2

Divide each term in $6 x = - 1$ by $6$ and simplify.

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Step 1.2.3.2.2.1

Divide each term in $6 x = - 1$ by $6$ .

$\frac{6 x}{6} = \frac{- 1}{6}$

Step 1.2.3.2.2.2

Simplify the left side.

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Step 1.2.3.2.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 1}{6}$

Step 1.2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{6}$

Step 1.2.3.2.2.3

Simplify the right side.

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Step 1.2.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{6}$

Step 1.2.4

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

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Step 1.2.4.1

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

$x^{2} + 1 = 0$

Step 1.2.4.2

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

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Step 1.2.4.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 1.2.4.2.2

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

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

Step 1.2.4.2.3

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

$x = \pm i$

Step 1.2.4.2.4

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

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Step 1.2.4.2.4.1

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

$x = i$

Step 1.2.4.2.4.2

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

$x = - i$

Step 1.2.4.2.4.3

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

$x = i, - i$

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

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

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Step 1.2.5.2.1

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

$x + 1 = 0$

Step 1.2.5.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.6

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

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

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

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

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Step 2.2.4.1

Multiply $6$ by $0$ .

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

Step 2.2.4.2

Add $0$ and $1$ .

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

Step 2.2.4.3

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

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

Step 2.2.4.4

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

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

Step 2.2.4.5

Add $0$ and $1$ .

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

Step 2.2.4.6

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

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

Step 2.2.4.7

Add $0$ and $1$ .

$y = 1^{2}$

Step 2.2.4.8

One to any power is one.

$y = 1$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4