Algebra Examples

$9 x^{2} - 18 x - 7 \div (3 x + 1)$

Step 1

Rewrite the division as a fraction.

$9 x^{2} - 18 x - \frac{7}{3 x + 1}$

Step 2

To write $9 x^{2}$ as a fraction with a common denominator, multiply by $\frac{3 x + 1}{3 x + 1}$ .

$- 18 x + \frac{9 x^{2} (3 x + 1)}{3 x + 1} - \frac{7}{3 x + 1}$

Step 3

Combine the numerators over the common denominator.

$- 18 x + \frac{9 x^{2} (3 x + 1) - 7}{3 x + 1}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$- 18 x + \frac{9 x^{2} (3 x) + 9 x^{2} \cdot 1 - 7}{3 x + 1}$

Step 4.2

Rewrite using the commutative property of multiplication.

$- 18 x + \frac{9 \cdot 3 x^{2} x + 9 x^{2} \cdot 1 - 7}{3 x + 1}$

Step 4.3

Multiply $9$ by $1$ .

$- 18 x + \frac{9 \cdot 3 x^{2} x + 9 x^{2} - 7}{3 x + 1}$

Step 4.4

Simplify each term.

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Step 4.4.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.4.1.1

Move $x$ .

$- 18 x + \frac{9 \cdot 3 (x \cdot x^{2}) + 9 x^{2} - 7}{3 x + 1}$

Step 4.4.1.2

Multiply $x$ by $x^{2}$ .

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Step 4.4.1.2.1

Raise $x$ to the power of $1$ .

$- 18 x + \frac{9 \cdot 3 (x^{1} x^{2}) + 9 x^{2} - 7}{3 x + 1}$

Step 4.4.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 18 x + \frac{9 \cdot 3 x^{1 + 2} + 9 x^{2} - 7}{3 x + 1}$

Step 4.4.1.3

Add $1$ and $2$ .

$- 18 x + \frac{9 \cdot 3 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 4.4.2

Multiply $9$ by $3$ .

$- 18 x + \frac{27 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 5

To write $- 18 x$ as a fraction with a common denominator, multiply by $\frac{3 x + 1}{3 x + 1}$ .

$- 18 x \cdot \frac{3 x + 1}{3 x + 1} + \frac{27 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 6

Combine $- 18 x$ and $\frac{3 x + 1}{3 x + 1}$ .

$\frac{- 18 x (3 x + 1)}{3 x + 1} + \frac{27 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 7

Combine the numerators over the common denominator.

$\frac{- 18 x (3 x + 1) + 27 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{- 18 x (3 x) - 18 x \cdot 1 + 27 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 8.2

Rewrite using the commutative property of multiplication.

$\frac{- 18 \cdot 3 x \cdot x - 18 x \cdot 1 + 27 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 8.3

Multiply $- 18$ by $1$ .

$\frac{- 18 \cdot 3 x \cdot x - 18 x + 27 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 8.4

Simplify each term.

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Step 8.4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 8.4.1.1

Move $x$ .

$\frac{- 18 \cdot 3 (x \cdot x) - 18 x + 27 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 8.4.1.2

Multiply $x$ by $x$ .

$\frac{- 18 \cdot 3 x^{2} - 18 x + 27 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 8.4.2

Multiply $- 18$ by $3$ .

$\frac{- 54 x^{2} - 18 x + 27 x^{3} + 9 x^{2} - 7}{3 x + 1}$

Step 8.5

Add $- 54 x^{2}$ and $9 x^{2}$ .

$\frac{- 45 x^{2} - 18 x + 27 x^{3} - 7}{3 x + 1}$

Step 8.6

Reorder terms.

$\frac{27 x^{3} - 45 x^{2} - 18 x - 7}{3 x + 1}$