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))
This problem involves simplifying a complex rational expression, which is a division of multiple fractions that are themselves polynomial expressions. Specifically, you are asked to simplify the given algebraic fraction by performing the appropriate arithmetic operations (subtraction, multiplication, and division) on the polynomials. The expression includes factoring polynomials, canceling common factors between the numerators and denominators, and reducing the expression to its simplest form. It involves recognizing patterns such as difference of squares, factoring trinomials, and canceling out terms that appear in both the top and bottom of the fraction.
To simplify a complex fraction, multiply by the reciprocal of the divisor.
Begin by simplifying the numerator.
Express
Apply the difference of squares formula,
Factor
Identify the numbers as
Factor the expression accordingly.
Next, simplify the denominator.
Extract
Factor out
Repeat the factoring process for
Factor the expression to
Cancel out common factors.
Eliminate the common
Remove the common
Discard the common
Multiply
Factor the numerator
Extract
Factor
Factor the denominator
Extract
Factor
Cancel out common factors.
Eliminate the common
Remove the common
Multiply
Cancel out the common
Reciprocal Multiplication: When dividing by a fraction, multiply by its reciprocal.
Difference of Squares: The formula
Factoring Quadratics: The AC method involves finding two numbers that multiply to the product of the coefficient of
Common Factor Cancellation: When a factor appears in both the numerator and denominator, it can be canceled out.
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