Simplify (9-6x+3x^2)/3

Solution:

Step:1 Extract the common factor of $3$ from each term in the numerator. $\frac{3\times 3-3\times 2x+3\times x^2}{3}$

Step:2 Rewrite $-6x$ as $-3\times 2x$. $\frac{3\times 3-3\times 2x+3\times x^2}{3}$

Step:3 Combine the terms with the common factor of $3$. $\frac{3(3-2x)+3\times x^2}{3}$

Step:4 Recognize that $3x^2$ can be written as $3\times x^2$. $\frac{3(3-2x)+3\times x^2}{3}$

Step:5 Factor out the $3$ from the entire numerator. $\frac{3(3-2x+x^2)}{3}$

Step:6 Eliminate the common factor in the numerator and denominator.

Step:6.1 Identify the common factor of $3$. $\frac{3(3-2x+x^2)}{3\times 1}$

Step:6.2 Remove the common factor of $3$. $\frac{\cancel{3}(3-2x+x^2)}{\cancel{3}\times 1}$

Step:6.3 Simplify the expression. $\frac{3-2x+x^2}{1}$

Step:6.4 Divide the polynomial by $1$. $3-2x+x^2$

Knowledge Notes:

The problem at hand involves simplifying a rational expression, which is a fraction where the numerator and the denominator are polynomials. The process of simplifying such expressions often involves factoring out common factors from the numerator and denominator and then canceling them out.

Here are the relevant knowledge points and detailed explanations:

1. Factoring: This is the process of breaking down an expression into a product of simpler expressions. In this case, we factor out the number $3$ from each term in the numerator.

2. Common Factors: These are factors that are present in both the numerator and the denominator of a fraction. Common factors can be canceled out to simplify the expression.

3. Simplifying Fractions: When a common factor is present in both the numerator and the denominator, it can be divided out of both, which simplifies the fraction.

4. Polynomial Division: When a polynomial is divided by the number $1$, the result is the polynomial itself, as dividing by $1$ does not change the value of an expression.

In the given solution, the process involves systematically factoring out the common factor of $3$ from each term in the numerator and then canceling this factor against the $3$ in the denominator. The final result is a simplified polynomial expression.