To determine the other linear factor of the trinomial [tex]\(2x^2 + 13x + 6\)[/tex], given that one factor is [tex]\(x + 6\)[/tex], we need to perform polynomial division.
### Step-by-Step Solution:
1. Identify the given polynomial and the known factor:
- [tex]\(2x^2 + 13x + 6\)[/tex] is the given polynomial.
- [tex]\(x + 6\)[/tex] is the known factor.
2. Set up the polynomial division:
We start with [tex]\(2x^2 + 13x + 6\)[/tex] and divide it by [tex]\(x + 6\)[/tex].
3. Perform the polynomial division:
- Divide the leading term of the polynomial [tex]\(2x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{2x^2}{x} = 2x
\][/tex]
- Multiply [tex]\(2x\)[/tex] by the divisor [tex]\(x + 6\)[/tex]:
[tex]\[
2x(x + 6) = 2x^2 + 12x
\][/tex]
- Subtract the result from the original polynomial:
[tex]\[
(2x^2 + 13x + 6) - (2x^2 + 12x) = 13x + 6 - 12x = x + 6
\][/tex]
- Divide the new leading term [tex]\(x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{x}{x} = 1
\][/tex]
- Multiply [tex]\(1\)[/tex] by the divisor [tex]\(x + 6\)[/tex]:
[tex]\[
1(x + 6) = x + 6
\][/tex]
- Subtract the result from the previous remainder:
[tex]\[
(x + 6) - (x + 6) = 0
\][/tex]
4. Combine the results of the division:
The quotient obtained from this division is [tex]\(2x + 1\)[/tex], and the remainder is [tex]\(0\)[/tex].
So, the polynomial division shows that:
[tex]\[
2x^2 + 13x + 6 = (x + 6)(2x + 1)
\][/tex]
Therefore, the other linear factor is [tex]\(2x + 1\)[/tex].
### The correct answer is:
[tex]\[2x + 1\][/tex]
