Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x (x - 1) (x^{2} + 4) (x - 1) {(4 x + 3)}^{2} {(x - 3)}^{2} = 0$ .

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

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 $0$ .

$x = 0$

$x - 1 = 0$

$x^{2} + 4 = 0$

$x - 1 = 0$

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

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

Step 1.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x - 1 = 0$

Step 1.2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.5

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

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Step 1.2.5.1

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

$x^{2} + 4 = 0$

Step 1.2.5.2

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

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Step 1.2.5.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 1.2.5.2.2

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

$x = \pm \sqrt{- 4}$

Step 1.2.5.2.3

Simplify $\pm \sqrt{- 4}$ .

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Step 1.2.5.2.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 1.2.5.2.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 1.2.5.2.3.3

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

$x = \pm i \cdot \sqrt{4}$

Step 1.2.5.2.3.4

Rewrite $4$ as $2^{2}$ .

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

Step 1.2.5.2.3.5

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

$x = \pm i \cdot 2$

Step 1.2.5.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 1.2.5.2.4

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

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Step 1.2.5.2.4.1

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

$x = 2 i$

Step 1.2.5.2.4.2

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

$x = - 2 i$

Step 1.2.5.2.4.3

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

$x = 2 i, - 2 i$

Step 1.2.6

Set ${(4 x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set ${(4 x + 3)}^{2}$ equal to $0$ .

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

Step 1.2.6.2

Solve ${(4 x + 3)}^{2} = 0$ for $x$ .

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Step 1.2.6.2.1

Set the $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Step 1.2.6.2.2

Solve for $x$ .

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Step 1.2.6.2.2.1

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Step 1.2.6.2.2.2

Divide each term in $4 x = - 3$ by $4$ and simplify.

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Step 1.2.6.2.2.2.1

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 1.2.6.2.2.2.2

Simplify the left side.

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Step 1.2.6.2.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.6.2.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 1.2.6.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Step 1.2.6.2.2.2.3

Simplify the right side.

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Step 1.2.6.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

Step 1.2.7

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set ${(x - 3)}^{2}$ equal to $0$ .

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

Step 1.2.7.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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Step 1.2.7.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 1.2.7.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.8

The final solution is all the values that make $x (x - 1) (x^{2} + 4) (x - 1) {(4 x + 3)}^{2} {(x - 3)}^{2} = 0$ true.

$x = 0, 1, 2 i, - 2 i, - \frac{3}{4}, 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (1, 0), (- \frac{3}{4}, 0), (3, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Remove parentheses.

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

Step 2.2.5

Remove parentheses.

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

Step 2.2.6

Simplify $(0) ((0) - 1) ({(0)}^{2} + 4) ((0) - 1) {(4 (0) + 3)}^{2} {((0) - 3)}^{2}$ .

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Step 2.2.6.1

Subtract $1$ from $0$ .

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

Step 2.2.6.2

Multiply $0$ by $- 1$ .

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

Step 2.2.6.3

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

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

Step 2.2.6.4

Add $0$ and $4$ .

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

Step 2.2.6.5

Multiply $0$ by $4$ .

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

Step 2.2.6.6

Subtract $1$ from $0$ .

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

Step 2.2.6.7

Multiply $0$ by $- 1$ .

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

Step 2.2.6.8

Multiply $4$ by $0$ .

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

Step 2.2.6.9

Add $0$ and $3$ .

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

Step 2.2.6.10

Raise $3$ to the power of $2$ .

$y = 0 \cdot 9 {((0) - 3)}^{2}$

Step 2.2.6.11

Multiply $0$ by $9$ .

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

Step 2.2.6.12

Subtract $3$ from $0$ .

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

Step 2.2.6.13

Raise $- 3$ to the power of $2$ .

$y = 0 \cdot 9$

Step 2.2.6.14

Multiply $0$ by $9$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (1, 0), (- \frac{3}{4}, 0), (3, 0)$

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

Step 4