Simplify (6x^2+30x+36)÷(x+3)

Solution:

Step 1:

Express the division as a fraction: $\frac{6x^2 + 30x + 36}{x + 3}$.

Step 2:

Commence simplification of the numerator.

Step 2.1:

Extract the common factor of $6$ from $6x^2 + 30x + 36$.

Step 2.1.1:

Remove $6$ from $6x^2$: $\frac{6(x^2) + 30x + 36}{x + 3}$.

Step 2.1.2:

Remove $6$ from $30x$: $\frac{6(x^2) + 6(5x) + 36}{x + 3}$.

Step 2.1.3:

Remove $6$ from $36$: $\frac{6x^2 + 6(5x) + 6 \cdot 6}{x + 3}$.

Step 2.1.4:

Extract $6$ from $6x^2 + 6(5x)$: $\frac{6(x^2 + 5x) + 6 \cdot 6}{x + 3}$.

Step 2.1.5:

Extract $6$ completely: $\frac{6(x^2 + 5x + 6)}{x + 3}$.

Step 2.2:

Factor the quadratic expression $x^2 + 5x + 6$ by applying the AC method.

Step 2.2.1:

Identify two integers whose product equals $c$ and sum equals $b$. Here, find integers with a product of $6$ and a sum of $5$: $2$ and $3$.

Step 2.2.2:

Write the factored form with the identified integers: $\frac{6((x + 2)(x + 3))}{x + 3}$.

Step 3:

Proceed to simplify the terms.

Step 3.1:

Eliminate the common factor $x + 3$.

Step 3.1.1:

Cancel out the common term: $\frac{6(x + 2)(\cancel{x + 3})}{\cancel{x + 3}}$.

Step 3.1.2:

Simplify the expression by dividing $6(x + 2)$ by $1$: $6(x + 2)$.

Step 3.2:

Apply the distributive property to expand: $6x + 6 \cdot 2$.

Step 3.3:

Complete the multiplication of $6$ by $2$: $6x + 12$.

Knowledge Notes:

To simplify the expression $(6x^2+30x+36)÷(x+3)$, we use several algebraic techniques:

1. Factoring: This involves finding a common factor in all terms of the expression. In this case, we can factor out a $6$ from the numerator.

2. Fraction Representation: Division of polynomials can be represented as a fraction where the numerator is the dividend and the denominator is the divisor.

3. AC Method: This is a factoring technique used to factor quadratic expressions. It involves finding two numbers that multiply to give the product of the coefficient of $x^2$ (A) and the constant term (C), and add up to the coefficient of $x$ (B).

4. Cancellation: In fractions, if the numerator and denominator share a common factor, it can be canceled out to simplify the expression.

5. Distributive Property: This property is used to multiply a single term by each term inside a set of parentheses. In the final step, we distribute the $6$ to both $x$ and $2$.

By applying these techniques in a structured step-by-step approach, we can simplify the given algebraic fraction to its simplest form.