Questions: Factor by grouping (sometimes called the ac-method). 3 x^2+11 x+6 First, choose a form with appropriate signs. Then, fill in the blanks with numbers to be used for grouping. Finally, show the factorization. Form: 3 x^2+ square x+ square x+6 3 x^2+ square x - square x+6 3 x^2- square x+ square x+6 3 x^2 square x- square x+6

Factor by grouping (sometimes called the ac-method).

3 x^2+11 x+6

First, choose a form with appropriate signs. Then, fill in the blanks with numbers to be used for grouping. Finally, show the factorization.

Form: 3 x^2+ square x+ square x+6 3 x^2+ square x - square x+6 3 x^2- square x+ square x+6 3 x^2 square x- square x+6

Transcript text: Factor by grouping (sometimes called the ac-method). \[ 3 x^{2}+11 x+6 \] First, choose a form with appropriate signs. Then, fill in the blanks with numbers to be used for grouping. Finally, show the factorization. Form: $3 x^{2}+$ $\square$ $x+$ $\square$ $x+6$ $3 x^{2}+$ $\square$ $x$ - $\square$ $x+6$ $3 x^{2}-$ $\square$ $x+$ $\square$ $x+6$ $3 x^{2}$ $\square$ $x-$ $\square$ $x+6$

Solution

Solution Steps

Step 1: Identify the Product and Sum

To factor the quadratic expression $3x^2 + 11x + 6$ by grouping, we first need to find two numbers that multiply to the product of the coefficient of $x^2$ (which is 3) and the constant term (which is 6). This product is $3 \times 6 = 18$. We also need these two numbers to add up to the coefficient of $x$, which is 11.

Step 2: Find the Numbers

The two numbers that multiply to 18 and add up to 11 are 9 and 2. Therefore, we can rewrite the middle term $11x$ as $9x + 2x$.

Step 3: Rewrite the Expression

Using the numbers found, rewrite the expression: \[ 3x^2 + 11x + 6 = 3x^2 + 9x + 2x + 6 \]

Step 4: Factor by Grouping

Group the terms in pairs and factor each group: \[ (3x^2 + 9x) + (2x + 6) \]

Factor out the greatest common factor from each group: \[ 3x(x + 3) + 2(x + 3) \]

Step 5: Factor Out the Common Binomial

Notice that $(x + 3)$ is a common factor: \[ (3x + 2)(x + 3) \]