Algebra Examples

$\frac{9}{20 m^{4} + 100 m^{3}} - \frac{2 m^{2}}{3 m^{2} + 16 m + 5}$

Step 1

Simplify each term.

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Step 1.1

Factor $20 m^{3}$ out of $20 m^{4} + 100 m^{3}$ .

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Step 1.1.1

Factor $20 m^{3}$ out of $20 m^{4}$ .

$\frac{9}{20 m^{3} (m) + 100 m^{3}} - \frac{2 m^{2}}{3 m^{2} + 16 m + 5}$

Step 1.1.2

Factor $20 m^{3}$ out of $100 m^{3}$ .

$\frac{9}{20 m^{3} (m) + 20 m^{3} (5)} - \frac{2 m^{2}}{3 m^{2} + 16 m + 5}$

Step 1.1.3

Factor $20 m^{3}$ out of $20 m^{3} (m) + 20 m^{3} (5)$ .

$\frac{9}{20 m^{3} (m + 5)} - \frac{2 m^{2}}{3 m^{2} + 16 m + 5}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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 $a \cdot c = 3 \cdot 5 = 15$ and whose sum is $b = 16$ .

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Step 1.2.1.1

Factor $16$ out of $16 m$ .

$\frac{9}{20 m^{3} (m + 5)} - \frac{2 m^{2}}{3 m^{2} + 16 (m) + 5}$

Step 1.2.1.2

Rewrite $16$ as $1$ plus $15$

$\frac{9}{20 m^{3} (m + 5)} - \frac{2 m^{2}}{3 m^{2} + (1 + 15) m + 5}$

Step 1.2.1.3

Apply the distributive property.

$\frac{9}{20 m^{3} (m + 5)} - \frac{2 m^{2}}{3 m^{2} + 1 m + 15 m + 5}$

Step 1.2.1.4

Multiply $m$ by $1$ .

$\frac{9}{20 m^{3} (m + 5)} - \frac{2 m^{2}}{3 m^{2} + m + 15 m + 5}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{9}{20 m^{3} (m + 5)} - \frac{2 m^{2}}{(3 m^{2} + m) + 15 m + 5}$

Step 1.2.2.2

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

$\frac{9}{20 m^{3} (m + 5)} - \frac{2 m^{2}}{m (3 m + 1) + 5 (3 m + 1)}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $3 m + 1$ .

$\frac{9}{20 m^{3} (m + 5)} - \frac{2 m^{2}}{(3 m + 1) (m + 5)}$

Step 2

To write $\frac{9}{20 m^{3} (m + 5)}$ as a fraction with a common denominator, multiply by $\frac{3 m + 1}{3 m + 1}$ .

$\frac{9}{20 m^{3} (m + 5)} \cdot \frac{3 m + 1}{3 m + 1} - \frac{2 m^{2}}{(3 m + 1) (m + 5)}$

Step 3

To write $- \frac{2 m^{2}}{(3 m + 1) (m + 5)}$ as a fraction with a common denominator, multiply by $\frac{20 m^{3}}{20 m^{3}}$ .

$\frac{9}{20 m^{3} (m + 5)} \cdot \frac{3 m + 1}{3 m + 1} - \frac{2 m^{2}}{(3 m + 1) (m + 5)} \cdot \frac{20 m^{3}}{20 m^{3}}$

Step 4

Write each expression with a common denominator of $20 (m + 5) (3 m + 1) m^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{9}{20 m^{3} (m + 5)}$ by $\frac{3 m + 1}{3 m + 1}$ .

$\frac{9 (3 m + 1)}{20 m^{3} (m + 5) (3 m + 1)} - \frac{2 m^{2}}{(3 m + 1) (m + 5)} \cdot \frac{20 m^{3}}{20 m^{3}}$

Step 4.2

Multiply $\frac{2 m^{2}}{(3 m + 1) (m + 5)}$ by $\frac{20 m^{3}}{20 m^{3}}$ .

$\frac{9 (3 m + 1)}{20 m^{3} (m + 5) (3 m + 1)} - \frac{2 m^{2} (20 m^{3})}{(3 m + 1) (m + 5) (20 m^{3})}$

Step 4.3

Reorder the factors of $20 m^{3} (m + 5) (3 m + 1)$ .

$\frac{9 (3 m + 1)}{20 m^{3} (3 m + 1) (m + 5)} - \frac{2 m^{2} (20 m^{3})}{(3 m + 1) (m + 5) (20 m^{3})}$

Step 4.4

Reorder the factors of $(3 m + 1) (m + 5) (20 m^{3})$ .

$\frac{9 (3 m + 1)}{20 m^{3} (3 m + 1) (m + 5)} - \frac{2 m^{2} (20 m^{3})}{20 m^{3} (3 m + 1) (m + 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{9 (3 m + 1) - 2 m^{2} (20 m^{3})}{20 m^{3} (3 m + 1) (m + 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{9 (3 m) + 9 \cdot 1 - 2 m^{2} (20 m^{3})}{20 m^{3} (3 m + 1) (m + 5)}$

Step 6.2

Multiply $3$ by $9$ .

$\frac{27 m + 9 \cdot 1 - 2 m^{2} (20 m^{3})}{20 m^{3} (3 m + 1) (m + 5)}$

Step 6.3

Multiply $9$ by $1$ .

$\frac{27 m + 9 - 2 m^{2} (20 m^{3})}{20 m^{3} (3 m + 1) (m + 5)}$

Step 6.4

Rewrite using the commutative property of multiplication.

$\frac{27 m + 9 - 2 \cdot 20 m^{2} m^{3}}{20 m^{3} (3 m + 1) (m + 5)}$

Step 6.5

Multiply $m^{2}$ by $m^{3}$ by adding the exponents.

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Step 6.5.1

Move $m^{3}$ .

$\frac{27 m + 9 - 2 \cdot 20 (m^{3} m^{2})}{20 m^{3} (3 m + 1) (m + 5)}$

Step 6.5.2

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

$\frac{27 m + 9 - 2 \cdot 20 m^{3 + 2}}{20 m^{3} (3 m + 1) (m + 5)}$

Step 6.5.3

Add $3$ and $2$ .

$\frac{27 m + 9 - 2 \cdot 20 m^{5}}{20 m^{3} (3 m + 1) (m + 5)}$

Step 6.6

Multiply $- 2$ by $20$ .

$\frac{27 m + 9 - 40 m^{5}}{20 m^{3} (3 m + 1) (m + 5)}$

Step 6.7

Reorder terms.

$\frac{- 40 m^{5} + 27 m + 9}{20 m^{3} (3 m + 1) (m + 5)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- 40 m^{5}$ .

$\frac{- (40 m^{5}) + 27 m + 9}{20 m^{3} (3 m + 1) (m + 5)}$

Step 7.2

Factor $- 1$ out of $27 m$ .

$\frac{- (40 m^{5}) - (- 27 m) + 9}{20 m^{3} (3 m + 1) (m + 5)}$

Step 7.3

Factor $- 1$ out of $- (40 m^{5}) - (- 27 m)$ .

$\frac{- (40 m^{5} - 27 m) + 9}{20 m^{3} (3 m + 1) (m + 5)}$

Step 7.4

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- (40 m^{5} - 27 m) - 1 (- 9)}{20 m^{3} (3 m + 1) (m + 5)}$

Step 7.5

Factor $- 1$ out of $- (40 m^{5} - 27 m) - 1 (- 9)$ .

$\frac{- (40 m^{5} - 27 m - 9)}{20 m^{3} (3 m + 1) (m + 5)}$

Step 7.6

Simplify the expression.

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Step 7.6.1

Rewrite $- (40 m^{5} - 27 m - 9)$ as $- 1 (40 m^{5} - 27 m - 9)$ .

$\frac{- 1 (40 m^{5} - 27 m - 9)}{20 m^{3} (3 m + 1) (m + 5)}$

Step 7.6.2

Move the negative in front of the fraction.

$- \frac{40 m^{5} - 27 m - 9}{20 m^{3} (3 m + 1) (m + 5)}$