Simplify (6x^5+18x^4+6)/(3x^2)

Solution:

Step 1: Identify and eliminate common factors between the numerator and the denominator.

Step 1.1: Extract the factor of 3 from the term $6x^5$.

$\frac{3(2x^5) + 18x^4 + 6}{3x^2}$

Step 1.2: Extract the factor of 3 from the term $18x^4$.

$\frac{3(2x^5) + 3(6x^4) + 6}{3x^2}$

Step 1.3: Combine the terms with the factor of 3.

$\frac{3(2x^5 + 6x^4) + 6}{3x^2}$

Step 1.4: Extract the factor of 3 from the constant term 6.

$\frac{3(2x^5 + 6x^4) + 3 \cdot 2}{3x^2}$

Step 1.5: Combine all terms under the factor of 3.

$\frac{3(2x^5 + 6x^4 + 2)}{3x^2}$

Step 1.6: Cancel out the common factor of 3.

Step 1.6.1: Factor out the 3 from the denominator $3x^2$.

$\frac{3(2x^5 + 6x^4 + 2)}{3(x^2)}$

Step 1.6.2: Eliminate the common factor of 3.

$\frac{\cancel{3}(2x^5 + 6x^4 + 2)}{\cancel{3}x^2}$

Step 1.6.3: Simplify the expression.

$\frac{2x^5 + 6x^4 + 2}{x^2}$

Step 2: Factor out common terms from the simplified numerator.

Step 2.1: Extract the factor of 2 from the term $2x^5$.

$\frac{2(x^5) + 6x^4 + 2}{x^2}$

Step 2.2: Extract the factor of 2 from the term $6x^4$.

$\frac{2(x^5) + 2(3x^4) + 2}{x^2}$

Step 2.3: Extract the factor of 2 from the constant term 2.

$\frac{2(x^5) + 2(3x^4) + 2(1)}{x^2}$

Step 2.4: Combine the terms with the factor of 2.

$\frac{2(x^5 + 3x^4) + 2(1)}{x^2}$

Step 2.5: Factor out the 2 from the entire numerator.

$\frac{2(x^5 + 3x^4 + 1)}{x^2}$

Knowledge Notes:

The problem-solving process involves simplifying a polynomial expression by dividing it by a monomial. The key steps include:

1. Factoring: This is the process of breaking down expressions into products of other expressions or factors. In this case, we factor out common factors from the terms in the numerator.

2. Simplifying Fractions: When we have a fraction, we can simplify it by canceling out common factors in the numerator and the denominator. This is based on the property that $\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$ when $c \neq 0$.

3. Polynomial Division: When dividing polynomials, if a term in the numerator and the denominator share a common factor, that factor can be divided out. This simplifies the polynomial to a more manageable form.

4. Exponent Rules: When dividing terms with the same base, we subtract the exponents, according to the rule $\frac{x^m}{x^n} = x^{m-n}$, where $x$ is the base and $m$ and $n$ are the exponents.

In the given solution, the process starts by factoring out the greatest common divisor (GCD) of the terms in the numerator, which is 3. After canceling out the common factor of 3 from both the numerator and the denominator, the expression is further simplified by factoring out the GCD of the remaining terms in the numerator, which is 2. The final expression is obtained by canceling out all common factors and simplifying the terms according to the rules of exponents.