Algebra Examples

$f (x) = \frac{x^{3} - 15 x^{2} + 71 x - 105}{24} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21)}{8} + \frac{- 4 x^{2} + 24 x - 20}{16}$

Step 1

Write $f (x) = \frac{x^{3} - 15 x^{2} + 71 x - 105}{24} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21)}{8} + \frac{- 4 x^{2} + 24 x - 20}{16}$ as an equation.

$y = \frac{x^{3} - 15 x^{2} + 71 x - 105}{24} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21)}{8} + \frac{- 4 x^{2} + 24 x - 20}{16}$

Step 2

Simplify $\frac{x^{3} - 15 x^{2} + 71 x - 105}{24} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21)}{8} + \frac{- 4 x^{2} + 24 x - 20}{16}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{x^{3} - 15 x^{2} + 71 x - 105}{24}$ by $\frac{2}{2}$ .

$y = \frac{x^{3} - 15 x^{2} + 71 x - 105}{24} \cdot \frac{2}{2} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21)}{8} + \frac{- 4 x^{2} + 24 x - 20}{16}$

Step 2.1.2

Multiply $\frac{x^{3} - 15 x^{2} + 71 x - 105}{24}$ by $\frac{2}{2}$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{24 \cdot 2} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21)}{8} + \frac{- 4 x^{2} + 24 x - 20}{16}$

Step 2.1.3

Multiply $\frac{- (x^{3} - 11 x^{2} + 31 x - 21)}{8}$ by $\frac{6}{6}$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{24 \cdot 2} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21)}{8} \cdot \frac{6}{6} + \frac{- 4 x^{2} + 24 x - 20}{16}$

Step 2.1.4

Multiply $\frac{- (x^{3} - 11 x^{2} + 31 x - 21)}{8}$ by $\frac{6}{6}$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{24 \cdot 2} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6}{8 \cdot 6} + \frac{- 4 x^{2} + 24 x - 20}{16}$

Step 2.1.5

Multiply $\frac{- 4 x^{2} + 24 x - 20}{16}$ by $\frac{3}{3}$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{24 \cdot 2} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6}{8 \cdot 6} + \frac{- 4 x^{2} + 24 x - 20}{16} \cdot \frac{3}{3}$

Step 2.1.6

Multiply $\frac{- 4 x^{2} + 24 x - 20}{16}$ by $\frac{3}{3}$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{24 \cdot 2} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6}{8 \cdot 6} + \frac{(- 4 x^{2} + 24 x - 20) \cdot 3}{16 \cdot 3}$

Step 2.1.7

Reorder the factors of $24 \cdot 2$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{2 \cdot 24} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6}{8 \cdot 6} + \frac{(- 4 x^{2} + 24 x - 20) \cdot 3}{16 \cdot 3}$

Step 2.1.8

Multiply $2$ by $24$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{48} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6}{8 \cdot 6} + \frac{(- 4 x^{2} + 24 x - 20) \cdot 3}{16 \cdot 3}$

Step 2.1.9

Reorder the factors of $8 \cdot 6$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{48} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6}{6 \cdot 8} + \frac{(- 4 x^{2} + 24 x - 20) \cdot 3}{16 \cdot 3}$

Step 2.1.10

Multiply $6$ by $8$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{48} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6}{48} + \frac{(- 4 x^{2} + 24 x - 20) \cdot 3}{16 \cdot 3}$

Step 2.1.11

Reorder the factors of $16 \cdot 3$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{48} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6}{48} + \frac{(- 4 x^{2} + 24 x - 20) \cdot 3}{3 \cdot 16}$

Step 2.1.12

Multiply $3$ by $16$ .

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2}{48} + \frac{- (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6}{48} + \frac{(- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.2

Combine the numerators over the common denominator.

$y = \frac{(x^{3} - 15 x^{2} + 71 x - 105) \cdot 2 - (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$y = \frac{x^{3} \cdot 2 - 15 x^{2} \cdot 2 + 71 x \cdot 2 - 105 \cdot 2 - (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Move $2$ to the left of $x^{3}$ .

$y = \frac{2 \cdot x^{3} - 15 x^{2} \cdot 2 + 71 x \cdot 2 - 105 \cdot 2 - (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.2.2

Multiply $2$ by $- 15$ .

$y = \frac{2 \cdot x^{3} - 30 x^{2} + 71 x \cdot 2 - 105 \cdot 2 - (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.2.3

Multiply $2$ by $71$ .

$y = \frac{2 \cdot x^{3} - 30 x^{2} + 142 x - 105 \cdot 2 - (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.2.4

Multiply $- 105$ by $2$ .

$y = \frac{2 \cdot x^{3} - 30 x^{2} + 142 x - 210 - (x^{3} - 11 x^{2} + 31 x - 21) \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.3

Apply the distributive property.

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 + (- x^{3} - (- 11 x^{2}) - (31 x) - - 21) \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Multiply $- 11$ by $- 1$ .

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 + (- x^{3} + 11 x^{2} - (31 x) - - 21) \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.4.2

Multiply $31$ by $- 1$ .

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 + (- x^{3} + 11 x^{2} - 31 x - - 21) \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.4.3

Multiply $- 1$ by $- 21$ .

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 + (- x^{3} + 11 x^{2} - 31 x + 21) \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.5

Apply the distributive property.

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 - x^{3} \cdot 6 + 11 x^{2} \cdot 6 - 31 x \cdot 6 + 21 \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Multiply $6$ by $- 1$ .

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 - 6 x^{3} + 11 x^{2} \cdot 6 - 31 x \cdot 6 + 21 \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.6.2

Multiply $6$ by $11$ .

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 - 6 x^{3} + 66 x^{2} - 31 x \cdot 6 + 21 \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.6.3

Multiply $6$ by $- 31$ .

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 - 6 x^{3} + 66 x^{2} - 186 x + 21 \cdot 6 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.6.4

Multiply $21$ by $6$ .

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 - 6 x^{3} + 66 x^{2} - 186 x + 126 + (- 4 x^{2} + 24 x - 20) \cdot 3}{48}$

Step 2.3.7

Apply the distributive property.

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 - 6 x^{3} + 66 x^{2} - 186 x + 126 - 4 x^{2} \cdot 3 + 24 x \cdot 3 - 20 \cdot 3}{48}$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Multiply $3$ by $- 4$ .

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 - 6 x^{3} + 66 x^{2} - 186 x + 126 - 12 x^{2} + 24 x \cdot 3 - 20 \cdot 3}{48}$

Step 2.3.8.2

Multiply $3$ by $24$ .

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 - 6 x^{3} + 66 x^{2} - 186 x + 126 - 12 x^{2} + 72 x - 20 \cdot 3}{48}$

Step 2.3.8.3

Multiply $- 20$ by $3$ .

$y = \frac{2 x^{3} - 30 x^{2} + 142 x - 210 - 6 x^{3} + 66 x^{2} - 186 x + 126 - 12 x^{2} + 72 x - 60}{48}$

Step 2.4

Simplify terms.

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Step 2.4.1

Subtract $6 x^{3}$ from $2 x^{3}$ .

$y = \frac{- 4 x^{3} - 30 x^{2} + 142 x - 210 + 66 x^{2} - 186 x + 126 - 12 x^{2} + 72 x - 60}{48}$

Step 2.4.2

Add $- 30 x^{2}$ and $66 x^{2}$ .

$y = \frac{- 4 x^{3} + 36 x^{2} + 142 x - 210 - 186 x + 126 - 12 x^{2} + 72 x - 60}{48}$

Step 2.4.3

Subtract $12 x^{2}$ from $36 x^{2}$ .

$y = \frac{- 4 x^{3} + 24 x^{2} + 142 x - 210 - 186 x + 126 + 72 x - 60}{48}$

Step 2.4.4

Subtract $186 x$ from $142 x$ .

$y = \frac{- 4 x^{3} + 24 x^{2} - 44 x - 210 + 126 + 72 x - 60}{48}$

Step 2.4.5

Add $- 44 x$ and $72 x$ .

$y = \frac{- 4 x^{3} + 24 x^{2} + 28 x - 210 + 126 - 60}{48}$

Step 2.4.6

Simplify by adding and subtracting.

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Step 2.4.6.1

Add $- 210$ and $126$ .

$y = \frac{- 4 x^{3} + 24 x^{2} + 28 x - 84 - 60}{48}$

Step 2.4.6.2

Subtract $60$ from $- 84$ .

$y = \frac{- 4 x^{3} + 24 x^{2} + 28 x - 144}{48}$

Step 2.4.7

Cancel the common factor of $- 4 x^{3} + 24 x^{2} + 28 x - 144$ and $48$ .

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Step 2.4.7.1

Factor $4$ out of $- 4 x^{3}$ .

$y = \frac{4 (- x^{3}) + 24 x^{2} + 28 x - 144}{48}$

Step 2.4.7.2

Factor $4$ out of $24 x^{2}$ .

$y = \frac{4 (- x^{3}) + 4 (6 x^{2}) + 28 x - 144}{48}$

Step 2.4.7.3

Factor $4$ out of $4 (- x^{3}) + 4 (6 x^{2})$ .

$y = \frac{4 (- x^{3} + 6 x^{2}) + 28 x - 144}{48}$

Step 2.4.7.4

Factor $4$ out of $28 x$ .

$y = \frac{4 (- x^{3} + 6 x^{2}) + 4 (7 x) - 144}{48}$

Step 2.4.7.5

Factor $4$ out of $4 (- x^{3} + 6 x^{2}) + 4 (7 x)$ .

$y = \frac{4 (- x^{3} + 6 x^{2} + 7 x) - 144}{48}$

Step 2.4.7.6

Factor $4$ out of $- 144$ .

$y = \frac{4 (- x^{3} + 6 x^{2} + 7 x) + 4 (- 36)}{48}$

Step 2.4.7.7

Factor $4$ out of $4 (- x^{3} + 6 x^{2} + 7 x) + 4 (- 36)$ .

$y = \frac{4 (- x^{3} + 6 x^{2} + 7 x - 36)}{48}$

Step 2.4.7.8

Cancel the common factors.

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Step 2.4.7.8.1

Factor $4$ out of $48$ .

$y = \frac{4 (- x^{3} + 6 x^{2} + 7 x - 36)}{4 (12)}$

Step 2.4.7.8.2

Cancel the common factor.

$y = \frac{4 (- x^{3} + 6 x^{2} + 7 x - 36)}{4 \cdot 12}$

Step 2.4.7.8.3

Rewrite the expression.

$y = \frac{- x^{3} + 6 x^{2} + 7 x - 36}{12}$

Step 2.4.8

Factor $- 1$ out of $- x^{3}$ .

$y = \frac{- (x^{3}) + 6 x^{2} + 7 x - 36}{12}$

Step 2.4.9

Factor $- 1$ out of $6 x^{2}$ .

$y = \frac{- (x^{3}) - (- 6 x^{2}) + 7 x - 36}{12}$

Step 2.4.10

Factor $- 1$ out of $- (x^{3}) - (- 6 x^{2})$ .

$y = \frac{- (x^{3} - 6 x^{2}) + 7 x - 36}{12}$

Step 2.4.11

Factor $- 1$ out of $7 x$ .

$y = \frac{- (x^{3} - 6 x^{2}) - (- 7 x) - 36}{12}$

Step 2.4.12

Factor $- 1$ out of $- (x^{3} - 6 x^{2}) - (- 7 x)$ .

$y = \frac{- (x^{3} - 6 x^{2} - 7 x) - 36}{12}$

Step 2.4.13

Rewrite $- 36$ as $- 1 (36)$ .

$y = \frac{- (x^{3} - 6 x^{2} - 7 x) - 1 (36)}{12}$

Step 2.4.14

Factor $- 1$ out of $- (x^{3} - 6 x^{2} - 7 x) - 1 (36)$ .

$y = \frac{- (x^{3} - 6 x^{2} - 7 x + 36)}{12}$

Step 2.4.15

Simplify the expression.

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Step 2.4.15.1

Rewrite $- (x^{3} - 6 x^{2} - 7 x + 36)$ as $- 1 (x^{3} - 6 x^{2} - 7 x + 36)$ .

$y = \frac{- 1 (x^{3} - 6 x^{2} - 7 x + 36)}{12}$

Step 2.4.15.2

Move the negative in front of the fraction.

$y = - \frac{x^{3} - 6 x^{2} - 7 x + 36}{12}$