Simplify ((x^2+3x-28)/(2x^3))÷((x^2-7x+12)/(x-3))

$\frac{x^{2} + 3 x - 28}{2 x^{3}} \div \frac{x^{2} - 7 x + 12}{x - 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 3 x - 28}{2 x^{3}} \cdot \frac{x - 3}{x^{2} - 7 x + 12}$

Step 2

Factor $x^{2} + 3 x - 28$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 28$ and whose sum is $3$ .

$- 4, 7$

Write the factored form using these integers.

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

Step 3

Factor $x^{2} - 7 x + 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 7$ .

$- 4, - 3$

Write the factored form using these integers.

$\frac{(x - 4) (x + 7)}{2 x^{3}} \cdot \frac{x - 3}{(x - 4) (x - 3)}$

Step 4

Simplify terms.

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$\frac{(x - 4) (x + 7)}{2 x^{3}} \cdot \frac{x - 3}{(x - 4) (x - 3)}$

Rewrite the expression.

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

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

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 7}{2 x^{3}}$

Simplify ((x^2+3x-28)/(2x^3))÷((x^2-7x+12)/(x-3))

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