Simplify (48x^11-12x^10+30x^8+12x^6+5x^5)/(6x^8)

Solution:

Step 1: Extract the common factor

Extract $x^5$ from each term in the numerator $48x^{11}-12x^{10}+30x^8+12x^6+5x^5$.

Step 1.1: Factor out from the first term

Extract $x^5$ from $48x^{11}$ to get $\frac{x^5(48x^6)-12x^{10}+30x^8+12x^6+5x^5}{6x^8}$.

Step 1.2: Factor out from the second term

Extract $x^5$ from $-12x^{10}$ to get $\frac{x^5(48x^6)+x^5(-12x^5)+30x^8+12x^6+5x^5}{6x^8}$.

Step 1.3: Factor out from the third term

Extract $x^5$ from $30x^8$ to get $\frac{x^5(48x^6)+x^5(-12x^5)+x^5(30x^3)+12x^6+5x^5}{6x^8}$.

Step 1.4: Factor out from the fourth term

Extract $x^5$ from $12x^6$ to get $\frac{x^5(48x^6)+x^5(-12x^5)+x^5(30x^3)+x^5(12x)+5x^5}{6x^8}$.

Step 1.5: Factor out from the fifth term

Extract $x^5$ from $5x^5$ to get $\frac{x^5(48x^6)+x^5(-12x^5)+x^5(30x^3)+x^5(12x)+x^5\cdot 5}{6x^8}$.

Step 1.6: Combine the factored terms

Combine the terms factored by $x^5$ to get $\frac{x^5(48x^6-12x^5+30x^3+12x+5)}{6x^8}$.

Step 2: Simplify by canceling common factors

Step 2.1: Factor out from the denominator

Factor $x^5$ out of $6x^8$ to get $\frac{x^5(48x^6-12x^5+30x^3+12x+5)}{x^5(6x^3)}$.

Step 2.2: Cancel the common factor

Cancel out the common $x^5$ factor to get $\frac{\cancel{x^5}(48x^6-12x^5+30x^3+12x+5)}{\cancel{x^5}(6x^3)}$.

Step 2.3: Rewrite the simplified expression

Rewrite the expression to get $\frac{48x^6-12x^5+30x^3+12x+5}{6x^3}$.

Knowledge Notes:

1. Factoring: The process of breaking down an expression into products of other expressions or factors. In this case, we factored out $x^5$ from each term in the numerator.

2. Common Factor: A term that is present in all parts of an expression. Here, $x^5$ is a common factor in the numerator and $x^5$ is also in the denominator.

3. Simplifying Fractions: The process of reducing fractions to their simplest form by canceling out common factors in the numerator and denominator.

4. Polynomial Division: When dividing polynomials, one can cancel out common factors, which is a form of simplifying the expression.

5. LaTeX Formatting: A typesetting system used to create formatted mathematical expressions. In this solution, LaTeX is used to properly display the mathematical expressions and operations.