Find the x and y Intercepts f(x)=x^3-4x^2-25x+100

$f (x) = x^{3} - 4 x^{2} - 25 x + 100$

Step 1

Find the x-intercepts.

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

$0 = x^{3} - 4 x^{2} - 25 x + 100$

Solve the equation.

Rewrite the equation as $x^{3} - 4 x^{2} - 25 x + 100 = 0$ .

$x^{3} - 4 x^{2} - 25 x + 100 = 0$

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{3} - 4 x^{2}) - 25 x + 100 = 0$

Factor out the greatest common factor (GCF) from each group.

$x^{2} (x - 4) - 25 (x - 4) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 4$ .

$(x - 4) (x^{2} - 25) = 0$

Rewrite $25$ as $5^{2}$ .

$(x - 4) (x^{2} - 5^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$(x - 4) ((x + 5) (x - 5)) = 0$

Remove unnecessary parentheses.

$(x - 4) (x + 5) (x - 5) = 0$

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

$x - 4 = 0$

$x + 5 = 0$

$x - 5 = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $(x - 4) (x + 5) (x - 5) = 0$ true.

$x = 4, - 5, 5$

x-intercept(s) in point form.

x-intercept(s): $(4, 0), (- 5, 0), (5, 0)$

Step 2

Find the y-intercepts.

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

$y = {(0)}^{3} - 4 {(0)}^{2} - 25 \cdot 0 + 100$

Solve the equation.

Remove parentheses.

$y = 0^{3} - 4 {(0)}^{2} - 25 \cdot 0 + 100$

Remove parentheses.

$y = 0^{3} - 4 \cdot 0^{2} - 25 \cdot 0 + 100$

Remove parentheses.

$y = {(0)}^{3} - 4 {(0)}^{2} - 25 \cdot 0 + 100$

Simplify ${(0)}^{3} - 4 {(0)}^{2} - 25 \cdot 0 + 100$ .

Simplify each term.

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

$y = 0 - 4 {(0)}^{2} - 25 \cdot 0 + 100$

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

$y = 0 - 4 \cdot 0 - 25 \cdot 0 + 100$

Multiply $- 4$ by $0$ .

$y = 0 + 0 - 25 \cdot 0 + 100$

Multiply $- 25$ by $0$ .

$y = 0 + 0 + 0 + 100$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0 + 100$

Add $0$ and $0$ .

$y = 0 + 100$

Add $0$ and $100$ .

$y = 100$

y-intercept(s) in point form.

y-intercept(s): $(0, 100)$

Step 3

List the intersections.

x-intercept(s): $(4, 0), (- 5, 0), (5, 0)$

y-intercept(s): $(0, 100)$

Step 4

Find the x and y Intercepts f(x)=x^3-4x^2-25x+100

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