Algebra Examples

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

Step 1

Factor $3 (x - 2) (x^{2} - 1) (2 x - 3) {(x + 2)}^{4}$ .

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Step 1.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2

Factor.

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Step 1.2.1

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 $b = 1$ .

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

Step 1.2.2

Remove unnecessary parentheses.

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

Step 2

Cancel the common factor of $x + 2$ and ${(x + 2)}^{4}$ .

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Step 2.1

Factor $x + 2$ out of $4 (x + 2) (x + 1) (3 x + 4) (2 x - 3)$ .

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $x + 2$ out of $3 (x - 2) (x + 1) (x - 1) (2 x - 3) {(x + 2)}^{4}$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Cancel the common factor of $x + 1$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Cancel the common factor of $2 x - 3$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Multiply $4$ by $3 x + 4$ .

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

Step 6

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

$x + 1, 2 x - 3$

Step 7

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{4 (3 x + 4)}{(3 (x - 2) (x - 1)) {(x + 2)}^{3}}$ .

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Step 7.1

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

$x + 1 = 0$

Step 7.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7.3

Substitute $- 1$ for $x$ in $\frac{4 (3 x + 4)}{(3 (x - 2) (x - 1)) {(x + 2)}^{3}}$ and simplify.

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Step 7.3.1

Substitute $- 1$ for $x$ to find the $y$ coordinate of the hole.

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

Step 7.3.2

Simplify.

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Step 7.3.2.1

Simplify the numerator.

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Step 7.3.2.1.1

Multiply $3$ by $- 1$ .

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

Step 7.3.2.1.2

Add $- 3$ and $4$ .

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

Step 7.3.2.2

Simplify the denominator.

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Step 7.3.2.2.1

Subtract $2$ from $- 1$ .

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

Step 7.3.2.2.2

Multiply $3$ by $- 3$ .

$\frac{4 \cdot 1}{- 9 (- 1 - 1) {(- 1 + 2)}^{3}}$

Step 7.3.2.2.3

Subtract $1$ from $- 1$ .

$\frac{4 \cdot 1}{- 9 \cdot - 2 {(- 1 + 2)}^{3}}$

Step 7.3.2.2.4

Multiply $- 9$ by $- 2$ .

$\frac{4 \cdot 1}{18 {(- 1 + 2)}^{3}}$

Step 7.3.2.2.5

Add $- 1$ and $2$ .

$\frac{4 \cdot 1}{18 \cdot 1^{3}}$

Step 7.3.2.2.6

One to any power is one.

$\frac{4 \cdot 1}{18 \cdot 1}$

Step 7.3.2.3

Multiply $4$ by $1$ .

$\frac{4}{18 \cdot 1}$

Step 7.3.2.4

Multiply $18$ by $1$ .

$\frac{4}{18}$

Step 7.3.2.5

Cancel the common factor of $4$ and $18$ .

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Step 7.3.2.5.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{18}$

Step 7.3.2.5.2

Cancel the common factors.

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Step 7.3.2.5.2.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 2}{2 \cdot 9}$

Step 7.3.2.5.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 9}$

Step 7.3.2.5.2.3

Rewrite the expression.

$\frac{2}{9}$

Step 7.4

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 7.5

Solve $2 x - 3 = 0$ for $x$ .

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Step 7.5.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 7.5.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 7.5.2.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Step 7.5.2.2

Simplify the left side.

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Step 7.5.2.2.1

Cancel the common factor of $2$ .

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Step 7.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Step 7.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 7.6

Substitute $\frac{3}{2}$ for $x$ in $\frac{4 (3 x + 4)}{(3 (x - 2) (x - 1)) {(x + 2)}^{3}}$ and simplify.

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Step 7.6.1

Substitute $\frac{3}{2}$ for $x$ to find the $y$ coordinate of the hole.

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

Step 7.6.2

Simplify.

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Step 7.6.2.1

Simplify the numerator.

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Step 7.6.2.1.1

Multiply $3 (\frac{3}{2})$ .

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Step 7.6.2.1.1.1

Combine $3$ and $\frac{3}{2}$ .

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

Step 7.6.2.1.1.2

Multiply $3$ by $3$ .

$\frac{4 (\frac{9}{2} + 4)}{3 (\frac{3}{2} - 2) (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 (\frac{9}{2} + 4 \cdot \frac{2}{2})}{3 (\frac{3}{2} - 2) (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.1.3

Combine $4$ and $\frac{2}{2}$ .

$\frac{4 (\frac{9}{2} + \frac{4 \cdot 2}{2})}{3 (\frac{3}{2} - 2) (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.1.4

Combine the numerators over the common denominator.

$\frac{4 \frac{9 + 4 \cdot 2}{2}}{3 (\frac{3}{2} - 2) (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.1.5

Simplify the numerator.

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Step 7.6.2.1.5.1

Multiply $4$ by $2$ .

$\frac{4 \frac{9 + 8}{2}}{3 (\frac{3}{2} - 2) (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.1.5.2

Add $9$ and $8$ .

$\frac{4 (\frac{17}{2})}{3 (\frac{3}{2} - 2) (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.2

Simplify the denominator.

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Step 7.6.2.2.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 (\frac{17}{2})}{3 (\frac{3}{2} - 2 \cdot \frac{2}{2}) (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.2.2

Combine $- 2$ and $\frac{2}{2}$ .

$\frac{4 (\frac{17}{2})}{3 (\frac{3}{2} + \frac{- 2 \cdot 2}{2}) (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.2.3

Combine the numerators over the common denominator.

$\frac{4 (\frac{17}{2})}{3 \frac{3 - 2 \cdot 2}{2} (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.2.4

Simplify the numerator.

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Step 7.6.2.2.4.1

Multiply $- 2$ by $2$ .

$\frac{4 (\frac{17}{2})}{3 \frac{3 - 4}{2} (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.2.4.2

Subtract $4$ from $3$ .

$\frac{4 (\frac{17}{2})}{3 (\frac{- 1}{2}) (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.2.5

Move the negative in front of the fraction.

$\frac{4 (\frac{17}{2})}{3 \cdot - 1 \frac{1}{2} (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.2.6

Combine exponents.

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Step 7.6.2.2.6.1

Multiply $3$ by $- 1$ .

$\frac{4 (\frac{17}{2})}{- 3 (\frac{1}{2}) (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.2.6.2

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{4 (\frac{17}{2})}{\frac{- 3}{2} (\frac{3}{2} - 1) {(\frac{3}{2} + 2)}^{3}}$

Step 7.6.2.2.6.3

Combine ${(\frac{3}{2} + 2)}^{3}$ and $\frac{- 3}{2}$ .

$\frac{4 (\frac{17}{2})}{\frac{{(\frac{3}{2} + 2)}^{3} \cdot - 3}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.7

Simplify the numerator.

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Step 7.6.2.2.7.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 (\frac{17}{2})}{\frac{{(\frac{3}{2} + 2 \cdot \frac{2}{2})}^{3} \cdot - 3}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.7.2

Combine $2$ and $\frac{2}{2}$ .

$\frac{4 (\frac{17}{2})}{\frac{{(\frac{3}{2} + \frac{2 \cdot 2}{2})}^{3} \cdot - 3}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.7.3

Combine the numerators over the common denominator.

$\frac{4 (\frac{17}{2})}{\frac{{(\frac{3 + 2 \cdot 2}{2})}^{3} \cdot - 3}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.7.4

Simplify the numerator.

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Step 7.6.2.2.7.4.1

Multiply $2$ by $2$ .

$\frac{4 (\frac{17}{2})}{\frac{{(\frac{3 + 4}{2})}^{3} \cdot - 3}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.7.4.2

Add $3$ and $4$ .

$\frac{4 (\frac{17}{2})}{\frac{{(\frac{7}{2})}^{3} \cdot - 3}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.7.5

Apply the product rule to $\frac{7}{2}$ .

$\frac{4 (\frac{17}{2})}{\frac{\frac{7^{3}}{2^{3}} \cdot - 3}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.7.6

Raise $7$ to the power of $3$ .

$\frac{4 (\frac{17}{2})}{\frac{\frac{343}{2^{3}} \cdot - 3}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.7.7

Raise $2$ to the power of $3$ .

$\frac{4 (\frac{17}{2})}{\frac{\frac{343}{8} \cdot - 3}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.8

Combine $\frac{343}{8}$ and $- 3$ .

$\frac{4 (\frac{17}{2})}{\frac{\frac{343 \cdot - 3}{8}}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.9

Multiply $343$ by $- 3$ .

$\frac{4 (\frac{17}{2})}{\frac{\frac{- 1029}{8}}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.10

Move the negative in front of the fraction.

$\frac{4 (\frac{17}{2})}{\frac{- \frac{1029}{8}}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.11

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 (\frac{17}{2})}{- \frac{1029}{8} \cdot \frac{1}{2} (\frac{3}{2} - 1)}$

Step 7.6.2.2.12

Multiply $- \frac{1029}{8} \cdot \frac{1}{2}$ .

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Step 7.6.2.2.12.1

Multiply $\frac{1}{2}$ by $\frac{1029}{8}$ .

$\frac{4 (\frac{17}{2})}{- \frac{1029}{2 \cdot 8} (\frac{3}{2} - 1)}$

Step 7.6.2.2.12.2

Multiply $2$ by $8$ .

$\frac{4 (\frac{17}{2})}{- \frac{1029}{16} (\frac{3}{2} - 1)}$

Step 7.6.2.2.13

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 (\frac{17}{2})}{- \frac{1029}{16} (\frac{3}{2} - 1 \cdot \frac{2}{2})}$

Step 7.6.2.2.14

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{4 (\frac{17}{2})}{- \frac{1029}{16} (\frac{3}{2} + \frac{- 1 \cdot 2}{2})}$

Step 7.6.2.2.15

Combine the numerators over the common denominator.

$\frac{4 (\frac{17}{2})}{- \frac{1029}{16} \cdot \frac{3 - 1 \cdot 2}{2}}$

Step 7.6.2.2.16

Simplify the numerator.

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Step 7.6.2.2.16.1

Multiply $- 1$ by $2$ .

$\frac{4 (\frac{17}{2})}{- \frac{1029}{16} \cdot \frac{3 - 2}{2}}$

Step 7.6.2.2.16.2

Subtract $2$ from $3$ .

$\frac{4 (\frac{17}{2})}{- \frac{1029}{16} \cdot \frac{1}{2}}$

Step 7.6.2.2.17

Combine exponents.

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Step 7.6.2.2.17.1

Multiply $\frac{1}{2}$ by $\frac{1029}{16}$ .

$\frac{4 (\frac{17}{2})}{- \frac{1029}{2 \cdot 16}}$

Step 7.6.2.2.17.2

Multiply $2$ by $16$ .

$\frac{4 (\frac{17}{2})}{- \frac{1029}{32}}$

Step 7.6.2.3

Combine $4$ and $\frac{17}{2}$ .

$\frac{\frac{4 \cdot 17}{2}}{- \frac{1029}{32}}$

Step 7.6.2.4

Multiply $4$ by $17$ .

$\frac{\frac{68}{2}}{- \frac{1029}{32}}$

Step 7.6.2.5

Divide $68$ by $2$ .

$\frac{34}{- \frac{1029}{32}}$

Step 7.6.2.6

Multiply the numerator by the reciprocal of the denominator.

$34 (- \frac{32}{1029})$

Step 7.6.2.7

Multiply $34 (- \frac{32}{1029})$ .

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Step 7.6.2.7.1

Multiply $- 1$ by $34$ .

$- 34 (\frac{32}{1029})$

Step 7.6.2.7.2

Combine $- 34$ and $\frac{32}{1029}$ .

$\frac{- 34 \cdot 32}{1029}$

Step 7.6.2.7.3

Multiply $- 34$ by $32$ .

$\frac{- 1088}{1029}$

Step 7.6.2.8

Move the negative in front of the fraction.

$- \frac{1088}{1029}$

Step 7.7

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

$(- 1, \frac{2}{9}), (\frac{3}{2}, - \frac{1088}{1029})$

Step 8