Find the x and y Intercepts f(x)=x(4x+1)(5x-4)(2x-5)

$f (x) = x (4 x + 1) (5 x - 4) (2 x - 5)$

Step 1

Find the x-intercepts.

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

$0 = x (4 x + 1) (5 x - 4) (2 x - 5)$

Solve the equation.

Rewrite the equation as $x (4 x + 1) (5 x - 4) (2 x - 5) = 0$ .

$x (4 x + 1) (5 x - 4) (2 x - 5) = 0$

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

$x = 0$

$4 x + 1 = 0$

$5 x - 4 = 0$

$2 x - 5 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $4 x + 1$ equal to $0$ and solve for $x$ .

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Solve $4 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Divide each term in $4 x = - 1$ by $4$ and simplify.

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Set $5 x - 4$ equal to $0$ and solve for $x$ .

Set $5 x - 4$ equal to $0$ .

$5 x - 4 = 0$

Solve $5 x - 4 = 0$ for $x$ .

Add $4$ to both sides of the equation.

$5 x = 4$

Divide each term in $5 x = 4$ by $5$ and simplify.

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Divide $x$ by $1$ .

$x = \frac{4}{5}$

Set $2 x - 5$ equal to $0$ and solve for $x$ .

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Solve $2 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$2 x = 5$

Divide each term in $2 x = 5$ by $2$ and simplify.

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Divide $x$ by $1$ .

$x = \frac{5}{2}$

The final solution is all the values that make $x (4 x + 1) (5 x - 4) (2 x - 5) = 0$ true.

$x = 0, - \frac{1}{4}, \frac{4}{5}, \frac{5}{2}$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- \frac{1}{4}, 0), (\frac{4}{5}, 0), (\frac{5}{2}, 0)$

Step 2

Find the y-intercepts.

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

$y = (0) (4 (0) + 1) (5 (0) - 4) (2 (0) - 5)$

Solve the equation.

Multiply $0$ by $4 (0) + 1$ .

$y = 0 (4 (0) + 1) (5 (0) - 4) (2 (0) - 5)$

Remove parentheses.

$y = (0) (4 (0) + 1) (5 (0) - 4) (2 (0) - 5)$

Simplify $(0) (4 (0) + 1) (5 (0) - 4) (2 (0) - 5)$ .

Multiply $4$ by $0$ .

$y = 0 (0 + 1) (5 (0) - 4) (2 (0) - 5)$

Add $0$ and $1$ .

$y = 0 \cdot 1 (5 (0) - 4) (2 (0) - 5)$

Multiply $0$ by $1$ .

$y = 0 (5 (0) - 4) (2 (0) - 5)$

Multiply $5$ by $0$ .

$y = 0 (0 - 4) (2 (0) - 5)$

Subtract $4$ from $0$ .

$y = 0 \cdot - 4 (2 (0) - 5)$

Multiply $0$ by $- 4$ .

$y = 0 (2 (0) - 5)$

Multiply $2$ by $0$ .

$y = 0 (0 - 5)$

Subtract $5$ from $0$ .

$y = 0 \cdot - 5$

Multiply $0$ by $- 5$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- \frac{1}{4}, 0), (\frac{4}{5}, 0), (\frac{5}{2}, 0)$

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

Step 4

Find the x and y Intercepts f(x)=x(4x+1)(5x-4)(2x-5)

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