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 $\frac{x-10}{2x}$.