Problem

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.

36x26xx22x48x32x248x÷2x3+14x2+12xx39x210x

Answer

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Solution:

Step:1

To simplify a complex fraction, multiply by the reciprocal of the divisor.36x26xx22x48x32x248xx39x210x2x3+14x2+12x

Step:2

Begin by simplifying the numerator.

Step:2.1

Express 36 as 62.62x26xx22x48x32x248xx39x210x2x3+14x2+12x

Step:2.2

Apply the difference of squares formula, a2b2=(a+b)(ab), where a=6 and b=x.(6+x)(6x)6xx22x48x32x248xx39x210x2x3+14x2+12x

Step:3

Factor x22x48 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.(6+x)(6x)6x(x8)(x+6)x32x248xx39x210x2x3+14x2+12x

Step:4

Next, simplify the denominator.

Step:4.1

Extract x from x32x248x.

Step:4.1.1 to Step:4.1.5

Factor out x from each term, resulting in x(x22x48).(6+x)(6x)6x(x8)(x+6)x(x22x48)x39x210x2x3+14x2+12x

Step:4.2

Repeat the factoring process for x22x48.

Step:4.2.1 to Step:4.2.2

Factor the expression to (x8)(x+6).(6+x)(6x)6x(x8)(x+6)x(x8)(x+6)x39x210x2x3+14x2+12x

Step:5

Cancel out common factors.

Step:5.1

Eliminate the common (6x) term.

Step:5.2

Remove the common (x8) term.

Step:5.3

Discard the common (x+6) term.

Step:5.4

Multiply (6+x) by the remaining fraction.(6+x)(x10)2x(x+6)

Step:6

Factor the numerator x39x210x.

Step:6.1

Extract x from each term, resulting in x(x29x10).

Step:6.2

Factor x29x10 into (x10)(x+1).(6+x)(x10)(x+1)2x(x+6)

Step:7

Factor the denominator 2x3+14x2+12x.

Step:7.1

Extract 2x from each term, resulting in 2x(x2+7x+6).

Step:7.2

Factor x2+7x+6 into (x+1)(x+6).(6+x)(x10)(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.(6+x)(x10)2x(x+6)

Step:8.4

Cancel out the common (x+6) term, simplifying the expression to x102x.

Knowledge Notes:

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

  2. Difference of Squares: The formula a2b2=(a+b)(ab) 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 x2 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 x102x.

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