Algebra Examples

$\frac{(\frac{2 x^{2} - 7 x + 3}{10 x^{3}})}{(\frac{3 - x}{5})}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x^{2} - 7 x + 3}{10 x^{3}} \cdot \frac{5}{3 - x}$

Step 2

Cancel the common factor of $5$ .

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Step 2.1

Factor $5$ out of $10 x^{3}$ .

$\frac{2 x^{2} - 7 x + 3}{5 (2 x^{3})} \cdot \frac{5}{3 - x}$

Step 2.2

Cancel the common factor.

$\frac{2 x^{2} - 7 x + 3}{5 (2 x^{3})} \cdot \frac{5}{3 - x}$

Step 2.3

Rewrite the expression.

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

Step 3

Multiply $\frac{2 x^{2} - 7 x + 3}{2 x^{3}}$ by $\frac{1}{3 - x}$ .

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

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

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Step 4.1.1

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

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

Step 4.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

$\frac{2 x^{2} + (- 1 - 6) x + 3}{2 x^{3} (3 - x)}$

Step 4.1.3

Apply the distributive property.

$\frac{2 x^{2} - 1 x - 6 x + 3}{2 x^{3} (3 - x)}$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 1 x) - 6 x + 3}{2 x^{3} (3 - x)}$

Step 4.2.2

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

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

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

Step 5

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

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Step 5.1

Factor $- 1$ out of $x$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Reorder terms.

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

Step 5.5

Cancel the common factor.

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

Step 5.6

Rewrite the expression.

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

Step 6

Simplify the expression.

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Step 6.1

Move $- 1$ to the left of $2 x - 1$ .

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

Step 6.2

Move the negative in front of the fraction.

$- \frac{2 x - 1}{2 x^{3}}$