Simplify ((36-x^2)/(6-x)*(x^2-2x-48)/(x^3-2x^2-48x))÷((2x^3+14x^2+12x)/(x^3-9x^2-10x))

Solution:

Step:1

To simplify a complex fraction, multiply by the reciprocal of the divisor.$\frac{36 - x^{2}}{6 - x} \cdot \frac{x^{2} - 2x - 48}{x^{3} - 2x^{2} - 48x} \cdot \frac{x^{3} - 9x^{2} - 10x}{2x^{3} + 14x^{2} + 12x}$

Step:2

Begin by simplifying the numerator.

Step:2.1

Express $36$ as $6^{2}$.$\frac{6^{2} - x^{2}}{6 - x} \cdot \frac{x^{2} - 2x - 48}{x^{3} - 2x^{2} - 48x} \cdot \frac{x^{3} - 9x^{2} - 10x}{2x^{3} + 14x^{2} + 12x}$

Step:2.2

Apply the difference of squares formula, $a^{2} - b^{2} = (a + b)(a - b)$, where $a = 6$ and $b = x$.$\frac{(6 + x)(6 - x)}{6 - x} \cdot \frac{x^{2} - 2x - 48}{x^{3} - 2x^{2} - 48x} \cdot \frac{x^{3} - 9x^{2} - 10x}{2x^{3} + 14x^{2} + 12x}$

Step:3

Factor $x^{2} - 2x - 48$ by finding two numbers that multiply to $-48$ and add to $-2$.

Step:3.1

Identify the numbers as $-8$ and $6$.

Step:3.2

Factor the expression accordingly.$\frac{(6 + x)(6 - x)}{6 - x} \cdot \frac{(x - 8)(x + 6)}{x^{3} - 2x^{2} - 48x} \cdot \frac{x^{3} - 9x^{2} - 10x}{2x^{3} + 14x^{2} + 12x}$

Step:4

Next, simplify the denominator.

Step:4.1

Extract $x$ from $x^{3} - 2x^{2} - 48x$.

Step:4.1.1 to Step:4.1.5

Factor out $x$ from each term, resulting in $x(x^{2} - 2x - 48)$.$\frac{(6 + x)(6 - x)}{6 - x} \cdot \frac{(x - 8)(x + 6)}{x(x^{2} - 2x - 48)} \cdot \frac{x^{3} - 9x^{2} - 10x}{2x^{3} + 14x^{2} + 12x}$

Step:4.2

Repeat the factoring process for $x^{2} - 2x - 48$.

Step:4.2.1 to Step:4.2.2

Factor the expression to $(x - 8)(x + 6)$.$\frac{(6 + x)(6 - x)}{6 - x} \cdot \frac{(x - 8)(x + 6)}{x(x - 8)(x + 6)} \cdot \frac{x^{3} - 9x^{2} - 10x}{2x^{3} + 14x^{2} + 12x}$

Step:5

Cancel out common factors.

Step:5.1

Eliminate the common $(6 - x)$ term.

Step:5.2

Remove the common $(x - 8)$ term.

Step:5.3

Discard the common $(x + 6)$ term.

Step:5.4

Multiply $(6 + x)$ by the remaining fraction.$\frac{(6 + x)(x - 10)}{2x(x + 6)}$

Step:6

Factor the numerator $x^{3} - 9x^{2} - 10x$.

Step:6.1

Extract $x$ from each term, resulting in $x(x^{2} - 9x - 10)$.

Step:6.2

Factor $x^{2} - 9x - 10$ into $(x - 10)(x + 1)$.$\frac{(6 + x)(x - 10)(x + 1)}{2x(x + 6)}$

Step:7

Factor the denominator $2x^{3} + 14x^{2} + 12x$.

Step:7.1

Extract $2x$ from each term, resulting in $2x(x^{2} + 7x + 6)$.

Step:7.2

Factor $x^{2} + 7x + 6$ into $(x + 1)(x + 6)$.$\frac{(6 + x)(x - 10)(x + 1)}{2x(x + 1)(x + 6)}$

Step:8

Cancel out common factors.

Step:8.1

Eliminate the common $x$ term.

Step:8.2

Remove the common $(x + 1)$ term.

Step:8.3

Multiply $(6 + x)$ by the simplified fraction.$\frac{(6 + x)(x - 10)}{2x(x + 6)}$

Step:8.4

Cancel out the common $(x + 6)$ term, simplifying the expression to $\frac{x - 10}{2x}$.

Knowledge Notes:

1. Reciprocal Multiplication: When dividing by a fraction, multiply by its reciprocal.

2. Difference of Squares: The formula $a^{2} - b^{2} = (a + b)(a - b)$ is used to factor expressions where two terms are perfect squares separated by a subtraction sign.

3. Factoring Quadratics: The AC method involves finding two numbers that multiply to the product of the coefficient of $x^{2}$ and the constant term (AC) and add to the coefficient of $x$ (B). These numbers are then used to factor the quadratic expression.

4. Common Factor Cancellation: When a factor appears in both the numerator and denominator, it can be canceled out.

5. Simplifying Complex Fractions: The process involves factoring and canceling common terms in the numerator and denominator to simplify the expression.

In this problem, we applied these principles to simplify a complex algebraic fraction by factoring and canceling common terms. The final simplified form is $\frac{x - 10}{2x}$.