Find the x and y Intercepts 3x^3+22x^2+7x

$3 x^{3} + 22 x^{2} + 7 x$

Step 1

Write $3 x^{3} + 22 x^{2} + 7 x$ as an equation.

$y = 3 x^{3} + 22 x^{2} + 7 x$

Step 2

Find the x-intercepts.

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

$0 = 3 x^{3} + 22 x^{2} + 7 x$

Solve the equation.

Rewrite the equation as $3 x^{3} + 22 x^{2} + 7 x = 0$ .

$3 x^{3} + 22 x^{2} + 7 x = 0$

Factor the left side of the equation.

Factor $x$ out of $3 x^{3} + 22 x^{2} + 7 x$ .

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

$x (3 x^{2}) + 22 x^{2} + 7 x = 0$

Factor $x$ out of $22 x^{2}$ .

$x (3 x^{2}) + x (22 x) + 7 x = 0$

Factor $x$ out of $7 x$ .

$x (3 x^{2}) + x (22 x) + x \cdot 7 = 0$

Factor $x$ out of $x (3 x^{2}) + x (22 x)$ .

$x (3 x^{2} + 22 x) + x \cdot 7 = 0$

Factor $x$ out of $x (3 x^{2} + 22 x) + x \cdot 7$ .

$x (3 x^{2} + 22 x + 7) = 0$

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 $a \cdot c = 3 \cdot 7 = 21$ and whose sum is $b = 22$ .

Factor $22$ out of $22 x$ .

$x (3 x^{2} + 22 (x) + 7) = 0$

Rewrite $22$ as $1$ plus $21$

$x (3 x^{2} + (1 + 21) x + 7) = 0$

Apply the distributive property.

$x (3 x^{2} + 1 x + 21 x + 7) = 0$

Multiply $x$ by $1$ .

$x (3 x^{2} + x + 21 x + 7) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x ((3 x^{2} + x) + 21 x + 7) = 0$

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

$x (x (3 x + 1) + 7 (3 x + 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$x ((3 x + 1) (x + 7)) = 0$

Remove unnecessary parentheses.

$x (3 x + 1) (x + 7) = 0$

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

$x = 0$

$3 x + 1 = 0$

$x + 7 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $3 x + 1$ equal to $0$ and solve for $x$ .

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Solve $3 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Divide each term in $3 x = - 1$ by $3$ and simplify.

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

The final solution is all the values that make $x (3 x + 1) (x + 7) = 0$ true.

$x = 0, - \frac{1}{3}, - 7$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- \frac{1}{3}, 0), (- 7, 0)$

Step 3

Find the y-intercepts.

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

$y = 3 {(0)}^{3} + 22 {(0)}^{2} + 7 (0)$

Solve the equation.

Remove parentheses.

$y = 3 \cdot 0^{3} + 22 {(0)}^{2} + 7 (0)$

Remove parentheses.

$y = 3 \cdot 0^{3} + 22 \cdot 0^{2} + 7 (0)$

Remove parentheses.

$y = 3 {(0)}^{3} + 22 {(0)}^{2} + 7 (0)$

Simplify $3 {(0)}^{3} + 22 {(0)}^{2} + 7 (0)$ .

Simplify each term.

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

$y = 3 \cdot 0 + 22 {(0)}^{2} + 7 (0)$

Multiply $3$ by $0$ .

$y = 0 + 22 {(0)}^{2} + 7 (0)$

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

$y = 0 + 22 \cdot 0 + 7 (0)$

Multiply $22$ by $0$ .

$y = 0 + 0 + 7 (0)$

Multiply $7$ 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 4

List the intersections.

x-intercept(s): $(0, 0), (- \frac{1}{3}, 0), (- 7, 0)$

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

Step 5

Find the x and y Intercepts 3x^3+22x^2+7x

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