Find the x and y Intercepts f(x)=-3(x^2-7)(x-1)(x-4)^2(x-1)(x^2+1)

$f (x) = - 3 (x^{2} - 7) (x - 1) {(x - 4)}^{2} (x - 1) (x^{2} + 1)$

Step 1

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 3 (x^{2} - 7) (x - 1) {(x - 4)}^{2} (x - 1) (x^{2} + 1)$

Solve the equation.

Rewrite the equation as $- 3 (x^{2} - 7) (x - 1) {(x - 4)}^{2} (x - 1) (x^{2} + 1) = 0$ .

$- 3 (x^{2} - 7) (x - 1) {(x - 4)}^{2} (x - 1) (x^{2} + 1) = 0$

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

$x^{2} - 7 = 0$

$x - 1 = 0$

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

$x - 1 = 0$

$x^{2} + 1 = 0$

Set $x^{2} - 7$ equal to $0$ and solve for $x$ .

Set $x^{2} - 7$ equal to $0$ .

$x^{2} - 7 = 0$

Solve $x^{2} - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$x^{2} = 7$

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

$x = \pm \sqrt{7}$

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

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

$x = \sqrt{7}$

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

$x = - \sqrt{7}$

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

$x = \sqrt{7}, - \sqrt{7}$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

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

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

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

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

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

$x^{2} + 1 = 0$

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

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

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

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

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

$x = \pm i$

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

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

$x = i$

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

$x = - i$

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

$x = i, - i$

The final solution is all the values that make $- 3 (x^{2} - 7) (x - 1) {(x - 4)}^{2} (x - 1) (x^{2} + 1) = 0$ true.

$x = \sqrt{7}, - \sqrt{7}, 1, 4, i, - i$

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{7}, 0), (- \sqrt{7}, 0), (1, 0), (4, 0)$

Step 2

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 3 ({(0)}^{2} - 7) ((0) - 1) {((0) - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Solve the equation.

Remove parentheses.

$y = - 3 (0^{2} - 7) ((0) - 1) {((0) - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Remove parentheses.

$y = - 3 (0^{2} - 7) (0 - 1) {((0) - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Remove parentheses.

$y = - 3 (0^{2} - 7) (0 - 1) {(0 - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Remove parentheses.

$y = - 3 (0^{2} - 7) (0 - 1) {(0 - 4)}^{2} (0 - 1) ({(0)}^{2} + 1)$

Remove parentheses.

$y = - 3 (0^{2} - 7) (0 - 1) {(0 - 4)}^{2} (0 - 1) (0^{2} + 1)$

Remove parentheses.

$y = - 3 ({(0)}^{2} - 7) ((0) - 1) {((0) - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Simplify $- 3 ({(0)}^{2} - 7) ((0) - 1) {((0) - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$ .

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

$y = - 3 (0 - 7) ((0) - 1) {((0) - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Subtract $7$ from $0$ .

$y = - 3 \cdot - 7 ((0) - 1) {((0) - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Multiply $- 3$ by $- 7$ .

$y = 21 ((0) - 1) {((0) - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Subtract $1$ from $0$ .

$y = 21 \cdot - 1 {((0) - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Multiply $21$ by $- 1$ .

$y = - 21 {((0) - 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Subtract $4$ from $0$ .

$y = - 21 {(- 4)}^{2} ((0) - 1) ({(0)}^{2} + 1)$

Raise $- 4$ to the power of $2$ .

$y = - 21 \cdot 16 ((0) - 1) ({(0)}^{2} + 1)$

Multiply $- 21$ by $16$ .

$y = - 336 ((0) - 1) ({(0)}^{2} + 1)$

Subtract $1$ from $0$ .

$y = - 336 \cdot - 1 ({(0)}^{2} + 1)$

Multiply $- 336$ by $- 1$ .

$y = 336 ({(0)}^{2} + 1)$

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

$y = 336 (0 + 1)$

Add $0$ and $1$ .

$y = 336 \cdot 1$

Multiply $336$ by $1$ .

$y = 336$

y-intercept(s) in point form.

y-intercept(s): $(0, 336)$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{7}, 0), (- \sqrt{7}, 0), (1, 0), (4, 0)$

y-intercept(s): $(0, 336)$

Step 4

Find the x and y Intercepts f(x)=-3(x^2-7)(x-1)(x-4)^2(x-1)(x^2+1)

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