Algebra Examples

$P (x) = (x - 5) (x - 3) (x + 2)$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = (x - 5) (x - 3) (x + 2)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(x - 5) (x - 3) (x + 2) = 0$ .

$(x - 5) (x - 3) (x + 2) = 0$

Step 1.2.2

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

$x - 5 = 0$

$x - 3 = 0$

$x + 2 = 0$

Step 1.2.3

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

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Step 1.2.3.1

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

$x - 5 = 0$

Step 1.2.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 1.2.4

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

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Step 1.2.4.1

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

$x - 3 = 0$

Step 1.2.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.5

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

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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

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

$x = 5, 3, - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(5, 0), (3, 0), (- 2, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = ((0) - 5) ((0) - 3) ((0) + 2)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = (0 - 5) ((0) - 3) ((0) + 2)$

Step 2.2.2

Remove parentheses.

$y = (0 - 5) (0 - 3) ((0) + 2)$

Step 2.2.3

Remove parentheses.

$y = (0 - 5) (0 - 3) (0 + 2)$

Step 2.2.4

Remove parentheses.

$y = ((0) - 5) ((0) - 3) ((0) + 2)$

Step 2.2.5

Simplify $((0) - 5) ((0) - 3) ((0) + 2)$ .

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Step 2.2.5.1

Subtract $5$ from $0$ .

$y = - 5 ((0) - 3) ((0) + 2)$

Step 2.2.5.2

Subtract $3$ from $0$ .

$y = - 5 \cdot - 3 ((0) + 2)$

Step 2.2.5.3

Multiply $- 5$ by $- 3$ .

$y = 15 ((0) + 2)$

Step 2.2.5.4

Add $0$ and $2$ .

$y = 15 \cdot 2$

Step 2.2.5.5

Multiply $15$ by $2$ .

$y = 30$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 30)$

Step 3

List the intersections.

x-intercept(s): $(5, 0), (3, 0), (- 2, 0)$

y-intercept(s): $(0, 30)$

Step 4