Precalculus Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

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 $0$ .

$x + 1 = 0$

$x - 6 = 0$

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

Step 1.2.3

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

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Step 1.2.3.1

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

$x + 1 = 0$

Step 1.2.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.4

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

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Step 1.2.4.1

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

$x - 6 = 0$

Step 1.2.4.2

Add $6$ to both sides of the equation.

$x = 6$

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

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.6

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

$x = - 1, 6, 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 1, 0), (6, 0), (1, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Remove parentheses.

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

Step 2.2.5

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

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Step 2.2.5.1

Add $0$ and $1$ .

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

Step 2.2.5.2

Multiply $(0) - 6$ by $1$ .

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

Step 2.2.5.3

Subtract $6$ from $0$ .

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

Step 2.2.5.4

Subtract $1$ from $0$ .

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

Step 2.2.5.5

Raise $- 1$ to the power of $2$ .

$y = - 6 \cdot 1$

Step 2.2.5.6

Multiply $- 6$ by $1$ .

$y = - 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 6)$

Step 3

List the intersections.

x-intercept(s): $(- 1, 0), (6, 0), (1, 0)$

y-intercept(s): $(0, - 6)$

Step 4