Find the x and y Intercepts f(x)=5x^5-17x^4+6x^3

$f (x) = 5 x^{5} - 17 x^{4} + 6 x^{3}$

Step 1

Find the x-intercepts.

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

$0 = 5 x^{5} - 17 x^{4} + 6 x^{3}$

Solve the equation.

Rewrite the equation as $5 x^{5} - 17 x^{4} + 6 x^{3} = 0$ .

$5 x^{5} - 17 x^{4} + 6 x^{3} = 0$

Factor the left side of the equation.

Factor $x^{3}$ out of $5 x^{5} - 17 x^{4} + 6 x^{3}$ .

Factor $x^{3}$ out of $5 x^{5}$ .

$x^{3} (5 x^{2}) - 17 x^{4} + 6 x^{3} = 0$

Factor $x^{3}$ out of $- 17 x^{4}$ .

$x^{3} (5 x^{2}) + x^{3} (- 17 x) + 6 x^{3} = 0$

Factor $x^{3}$ out of $6 x^{3}$ .

$x^{3} (5 x^{2}) + x^{3} (- 17 x) + x^{3} \cdot 6 = 0$

Factor $x^{3}$ out of $x^{3} (5 x^{2}) + x^{3} (- 17 x)$ .

$x^{3} (5 x^{2} - 17 x) + x^{3} \cdot 6 = 0$

Factor $x^{3}$ out of $x^{3} (5 x^{2} - 17 x) + x^{3} \cdot 6$ .

$x^{3} (5 x^{2} - 17 x + 6) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 6 = 30$ and whose sum is $b = - 17$ .

Factor $- 17$ out of $- 17 x$ .

$x^{3} (5 x^{2} - 17 x + 6) = 0$

Rewrite $- 17$ as $- 2$ plus $- 15$

$x^{3} (5 x^{2} + (- 2 - 15) x + 6) = 0$

Apply the distributive property.

$x^{3} (5 x^{2} - 2 x - 15 x + 6) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x^{3} ((5 x^{2} - 2 x) - 15 x + 6) = 0$

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

$x^{3} (x (5 x - 2) - 3 (5 x - 2)) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 x - 2$ .

$x^{3} ((5 x - 2) (x - 3)) = 0$

Remove unnecessary parentheses.

$x^{3} (5 x - 2) (x - 3) = 0$

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

$x^{3} = 0$

$5 x - 2 = 0$

$x - 3 = 0$

Set $x^{3}$ equal to $0$ and solve for $x$ .

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Solve $x^{3} = 0$ for $x$ .

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

$x = \sqrt[3]{0}$

Simplify $\sqrt[3]{0}$ .

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

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

$x = 0$

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

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

$5 x - 2 = 0$

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

Add $2$ to both sides of the equation.

$5 x = 2$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Set $x - 3$ equal to $0$ and solve for $x$ .

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $x^{3} (5 x - 2) (x - 3) = 0$ true.

$x = 0, \frac{2}{5}, 3$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{2}{5}, 0), (3, 0)$

Step 2

Find the y-intercepts.

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

$y = 5 {(0)}^{5} - 17 {(0)}^{4} + 6 {(0)}^{3}$

Solve the equation.

Remove parentheses.

$y = 5 \cdot 0^{5} - 17 {(0)}^{4} + 6 {(0)}^{3}$

Remove parentheses.

$y = 5 \cdot 0^{5} - 17 \cdot 0^{4} + 6 {(0)}^{3}$

Remove parentheses.

$y = 5 \cdot 0^{5} - 17 \cdot 0^{4} + 6 \cdot 0^{3}$

Remove parentheses.

$y = 5 {(0)}^{5} - 17 {(0)}^{4} + 6 {(0)}^{3}$

Simplify $5 {(0)}^{5} - 17 {(0)}^{4} + 6 {(0)}^{3}$ .

Simplify each term.

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

$y = 5 \cdot 0 - 17 {(0)}^{4} + 6 {(0)}^{3}$

Multiply $5$ by $0$ .

$y = 0 - 17 {(0)}^{4} + 6 {(0)}^{3}$

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

$y = 0 - 17 \cdot 0 + 6 {(0)}^{3}$

Multiply $- 17$ by $0$ .

$y = 0 + 0 + 6 {(0)}^{3}$

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

$y = 0 + 0 + 6 \cdot 0$

Multiply $6$ by $0$ .

$y = 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\frac{2}{5}, 0), (3, 0)$

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

Step 4

Find the x and y Intercepts f(x)=5x^5-17x^4+6x^3

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