Question

Follow these steps to factor 6x* + 13x + 5 using the grouping method. ax?+ bx+c
Step 1: Identify ac & b
Product of leading coefficient and constant term, ac
ac =
choose your answer...
く
Coefficient of x-term or middle term, b
b =
choose your answer...
く
Step 2: List factors of ac and their associated sums.
Factors of 30
Sum of factors
1&
type your answer...
type your answer...
type your answer...
& 15
17
3 &
type your answer...
type your answer...
type your answer...
&6
11
Step 3: Write the pair of factors of ac that have a sum of b.
The factors of
choose your answer...
く
have a sum of 13.
Step 4: Use the pair of factors you found to rewrite bx as a sum of two terms. 6x?+13x+5= 6x^+ choose your answer...
+ 5
Step 5: Separate the polynomial's terms into two groups.
(6x}+3x) +(10x+5)
Step 6: Factor a GCF out of each group.
GCF of (6x* + 3x) = type your answer...
GCF of (10x + 5) = type your answer...
Step 7: Write the factorization of 6x? + 13x + 5
(2x + 1) (3x+5)

Answer 1

Let's follow the steps to factor the polynomial \(6x^2 + 13x + 5\) using the grouping method.

### Step 1: Identify \(ac\) & \(b\)
- The polynomial is \(ax^2 + bx + c\), where \(a = 6\), \(b = 13\), and \(c = 5\).
- To compute \(ac\):
\[
ac = a \cdot c = 6 \cdot 5 = 30
\]
- The coefficient of the \(x\)-term (middle term) is:
\[
b = 13
\]

### Step 2: List factors of \(ac\) and their associated sums.
- The factors of \(30\) are:
- \(1\) and \(30\) (Sum: \(31\))
- \(2\) and \(15\) (Sum: \(17\))
- **\(3\) and \(10\) (Sum: \(13\))**
- \(5\) and \(6\) (Sum: \(11\))

### Step 3: Write the pair of factors of \(ac\) that have a sum of \(b\).
- The factors of \(30\) that have a sum of \(13\) are:
\[
3 \text{ and } 10
\]

### Step 4: Use the pair of factors you found to rewrite \(bx\) as a sum of two terms.
- Rewrite \(13x\) using the factors:
\[
6x^2 + 3x + 10x + 5
\]

### Step 5: Separate the polynomial's terms into two groups.
- Group the terms:
\[
(6x^2 + 3x) + (10x + 5)
\]

### Step 6: Factor a GCF out of each group.
- The GCF of \(6x^2 + 3x\) is \(3x\):
\[
3x(2x + 1)
\]
- The GCF of \(10x + 5\) is \(5\):
\[
5(2x + 1)
\]

### Step 7: Write the factorization of \(6x^2 + 13x + 5\).
- Combine the factored terms:
\[
3x(2x + 1) + 5(2x + 1) = (2x + 1)(3x + 5)
\]

Thus, the complete factorization of \(6x^2 + 13x + 5\) is:
\[
(2x + 1)(3x + 5)
\]