Algebra Examples

$\frac{- 3 x^{2} - 21 x - 36}{2 x^{2} + 6 x}$

Step 1

Factor $- 3 x^{2} - 21 x - 36$ .

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Step 1.1

Factor $3$ out of $- 3 x^{2} - 21 x - 36$ .

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Step 1.1.1

Factor $3$ out of $- 3 x^{2}$ .

$\frac{3 (- x^{2}) - 21 x - 36}{2 x^{2} + 6 x}$

Step 1.1.2

Factor $3$ out of $- 21 x$ .

$\frac{3 (- x^{2}) + 3 (- 7 x) - 36}{2 x^{2} + 6 x}$

Step 1.1.3

Factor $3$ out of $- 36$ .

$\frac{3 (- x^{2}) + 3 (- 7 x) + 3 (- 12)}{2 x^{2} + 6 x}$

Step 1.1.4

Factor $3$ out of $3 (- x^{2}) + 3 (- 7 x)$ .

$\frac{3 (- x^{2} - 7 x) + 3 (- 12)}{2 x^{2} + 6 x}$

Step 1.1.5

Factor $3$ out of $3 (- x^{2} - 7 x) + 3 (- 12)$ .

$\frac{3 (- x^{2} - 7 x - 12)}{2 x^{2} + 6 x}$

Step 1.2

Factor.

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Step 1.2.1

Factor by grouping.

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Step 1.2.1.1

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 $a \cdot c = - 1 \cdot - 12 = 12$ and whose sum is $b = - 7$ .

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Step 1.2.1.1.1

Factor $- 7$ out of $- 7 x$ .

$\frac{3 (- x^{2} - 7 (x) - 12)}{2 x^{2} + 6 x}$

Step 1.2.1.1.2

Rewrite $- 7$ as $- 3$ plus $- 4$

$\frac{3 (- x^{2} + (- 3 - 4) x - 12)}{2 x^{2} + 6 x}$

Step 1.2.1.1.3

Apply the distributive property.

$\frac{3 (- x^{2} - 3 x - 4 x - 12)}{2 x^{2} + 6 x}$

Step 1.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.1.2.1

Group the first two terms and the last two terms.

$\frac{3 ((- x^{2} - 3 x) - 4 x - 12)}{2 x^{2} + 6 x}$

Step 1.2.1.2.2

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

$\frac{3 (x (- x - 3) + 4 (- x - 3))}{2 x^{2} + 6 x}$

Step 1.2.1.3

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

$\frac{3 ((- x - 3) (x + 4))}{2 x^{2} + 6 x}$

Step 1.2.2

Remove unnecessary parentheses.

$\frac{3 (- x - 3) (x + 4)}{2 x^{2} + 6 x}$

Step 2

Factor $2 x$ out of $2 x^{2} + 6 x$ .

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Step 2.1

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

$\frac{3 (- x - 3) (x + 4)}{2 x (x) + 6 x}$

Step 2.2

Factor $2 x$ out of $6 x$ .

$\frac{3 (- x - 3) (x + 4)}{2 x (x) + 2 x (3)}$

Step 2.3

Factor $2 x$ out of $2 x (x) + 2 x (3)$ .

$\frac{3 (- x - 3) (x + 4)}{2 x (x + 3)}$

Step 3

Cancel the common factor of $- x - 3$ and $x + 3$ .

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Step 3.1

Factor $- 1$ out of $- x$ .

$\frac{3 (- (x) - 3) (x + 4)}{2 x (x + 3)}$

Step 3.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{3 (- (x) - 1 (3)) (x + 4)}{2 x (x + 3)}$

Step 3.3

Factor $- 1$ out of $- (x) - 1 (3)$ .

$\frac{3 (- (x + 3)) (x + 4)}{2 x (x + 3)}$

Step 3.4

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$\frac{3 (- 1 (x + 3)) (x + 4)}{2 x (x + 3)}$

Step 3.5

Cancel the common factor.

$\frac{3 (- 1 (x + 3)) (x + 4)}{2 x (x + 3)}$

Step 3.6

Rewrite the expression.

$\frac{3 \cdot (- 1) (x + 4)}{2 x}$

Step 4

Combine exponents.

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Step 4.1

Factor out negative.

$\frac{- (3 (x + 4))}{2 x}$

Step 4.2

Multiply $3$ by $- 1$ .

$\frac{- 3 (x + 4)}{2 x}$

Step 5

Move the negative in front of the fraction.

$- \frac{3 (x + 4)}{2 x}$

Step 6

To find the holes in the graph, look at the denominator factors that were cancelled.

$x + 3$

Step 7

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $- \frac{3 (x + 4)}{2 x}$ .

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Step 7.1

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

$x + 3 = 0$

Step 7.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 7.3

Substitute $- 3$ for $x$ in $- \frac{3 (x + 4)}{2 x}$ and simplify.

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Step 7.3.1

Substitute $- 3$ for $x$ to find the $y$ coordinate of the hole.

$- \frac{3 (- 3 + 4)}{2 \cdot - 3}$

Step 7.3.2

Simplify.

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Step 7.3.2.1

Cancel the common factors.

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Step 7.3.2.1.1

Factor $3$ out of $2 \cdot - 3$ .

$- \frac{3 (- 3 + 4)}{3 (2 \cdot - 1)}$

Step 7.3.2.1.2

Cancel the common factor.

$- \frac{3 (- 3 + 4)}{3 (2 \cdot - 1)}$

Step 7.3.2.1.3

Rewrite the expression.

$- \frac{- 3 + 4}{2 \cdot - 1}$

Step 7.3.2.2

Add $- 3$ and $4$ .

$- \frac{1}{2 \cdot - 1}$

Step 7.3.2.3

Multiply $2$ by $- 1$ .

$- \frac{1}{- 2}$

Step 7.3.2.4

Move the negative in front of the fraction.

$- - \frac{1}{2}$

Step 7.3.2.5

Multiply $- - \frac{1}{2}$ .

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Step 7.3.2.5.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2})$

Step 7.3.2.5.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$

Step 7.4

The holes in the graph are the points where any of the cancelled factors are equal to $0$ .

$(- 3, \frac{1}{2})$

Step 8