Finite Math Examples

$P (x) = x^{3} - 7 x^{2} - 4 x + 28$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

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

$0 = x^{3} - 7 x^{2} - 4 x + 28$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $x^{3} - 7 x^{2} - 4 x + 28 = 0$ .

$x^{3} - 7 x^{2} - 4 x + 28 = 0$

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.2.1.1

Group the first two terms and the last two terms.

$(x^{3} - 7 x^{2}) - 4 x + 28 = 0$

Step 1.2.2.1.2

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

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

Step 1.2.2.2

Factor the polynomial by factoring out the greatest common factor, $x - 7$ .

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

Step 1.2.2.3

Rewrite $4$ as $2^{2}$ .

$(x - 7) (x^{2} - 2^{2}) = 0$

Step 1.2.2.4

Factor.

Tap for more steps...

Step 1.2.2.4.1

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 $b = 2$ .

$(x - 7) ((x + 2) (x - 2)) = 0$

Step 1.2.2.4.2

Remove unnecessary parentheses.

$(x - 7) (x + 2) (x - 2) = 0$

Step 1.2.3

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

$x - 7 = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$x - 7 = 0$

Step 1.2.4.2

Add $7$ to both sides of the equation.

$x = 7$

Step 1.2.5

Set $x + 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 1.2.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.6

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

Tap for more steps...

Step 1.2.6.1

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

$x - 2 = 0$

Step 1.2.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.7

The final solution is all the values that make $(x - 7) (x + 2) (x - 2) = 0$ true.

$x = 7, - 2, 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(7, 0), (- 2, 0), (2, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

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

$y = {(0)}^{3} - 7 {(0)}^{2} - 4 \cdot 0 + 28$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = 0^{3} - 7 {(0)}^{2} - 4 \cdot 0 + 28$

Step 2.2.2

Remove parentheses.

$y = 0^{3} - 7 \cdot 0^{2} - 4 \cdot 0 + 28$

Step 2.2.3

Remove parentheses.

$y = {(0)}^{3} - 7 {(0)}^{2} - 4 \cdot 0 + 28$

Step 2.2.4

Simplify ${(0)}^{3} - 7 {(0)}^{2} - 4 \cdot 0 + 28$ .

Tap for more steps...

Step 2.2.4.1

Simplify each term.

Tap for more steps...

Step 2.2.4.1.1

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

$y = 0 - 7 {(0)}^{2} - 4 \cdot 0 + 28$

Step 2.2.4.1.2

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

$y = 0 - 7 \cdot 0 - 4 \cdot 0 + 28$

Step 2.2.4.1.3

Multiply $- 7$ by $0$ .

$y = 0 + 0 - 4 \cdot 0 + 28$

Step 2.2.4.1.4

Multiply $- 4$ by $0$ .

$y = 0 + 0 + 0 + 28$

Step 2.2.4.2

Simplify by adding numbers.

Tap for more steps...

Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 28$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 + 28$

Step 2.2.4.2.3

Add $0$ and $28$ .

$y = 28$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 28)$

Step 3

List the intersections.

x-intercept(s): $(7, 0), (- 2, 0), (2, 0)$

y-intercept(s): $(0, 28)$

Step 4