Simplify (6x^2+30x+36)÷(x+3)

The problem is asking to perform polynomial division, simplifying the expression by dividing the quadratic polynomial, 6x^2+30x+36, by the linear polynomial, x+3.

$(6 x^{2} + 30 x + 36) \div (x + 3)$

Answer

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Solution:

Step 1:

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

Step 2:

Commence simplification of the numerator.

Step 2.1:

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

Step 2.1.1:

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

Step 2.1.2:

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

Step 2.1.3:

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

Step 2.1.4:

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

Step 2.1.5:

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

Step 2.2:

Factor the quadratic expression $x^{2} + 5 x + 6$ 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: $\frac{6 (x + 2) (x + 3)}{x + 3}$ .

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: $6 x + 6 \cdot 2$ .

Step 3.3:

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

Knowledge Notes:

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

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.
Fraction Representation: Division of polynomials can be represented as a fraction where the numerator is the dividend and the denominator is the divisor.
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).
Cancellation: In fractions, if the numerator and denominator share a common factor, it can be canceled out to simplify the expression.
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.