To find the other linear factor of the trinomial [tex]\(2x^2 + 13x + 6\)[/tex], given that one of the factors is [tex]\(x + 6\)[/tex], we can proceed with polynomial division or factorization techniques. Here's the step-by-step process to determine the other factor:
1. Given Information:
We know that the trinomial [tex]\(2x^2 + 13x + 6\)[/tex] can be factored as [tex]\((x + 6)(\text{Other Factor})\)[/tex].
2. Setup the Solution:
Let the other factor be of the form [tex]\((ax + b)\)[/tex]. Therefore, we can express the trinomial as:
[tex]\[
(x + 6)(ax + b) = 2x^2 + 13x + 6
\][/tex]
3. Expand the Product:
Expanding the left side, we get:
[tex]\[
(x + 6)(ax + b) = ax^2 + bx + 6ax + 6b
\][/tex]
4. Combine Like Terms:
Combine like terms in the expanded form:
[tex]\[
ax^2 + (b + 6a)x + 6b
\][/tex]
5. Match Coefficients:
Now, we compare the coefficients of corresponding terms from the expanded form to the original trinomial [tex]\(2x^2 + 13x + 6\)[/tex]:
- The coefficient of [tex]\(x^2\)[/tex] gives us [tex]\(a = 2\)[/tex].
- The coefficient of [tex]\(x\)[/tex] gives us [tex]\(b + 6a = 13\)[/tex].
- The constant term gives us [tex]\(6b = 6\)[/tex].
6. Solve the System of Equations:
[tex]\[
a = 2
\][/tex]
Substitute [tex]\(a = 2\)[/tex] into [tex]\(b + 6a = 13\)[/tex]:
[tex]\[
b + 6(2) = 13 \implies b + 12 = 13 \implies b = 1
\][/tex]
For the constant term, verify [tex]\(6b = 6\)[/tex]:
[tex]\[
6(1) = 6
\][/tex]
7. Determine the Other Factor:
Now, we have [tex]\(a = 2\)[/tex] and [tex]\(b = 1\)[/tex], so the other linear factor is:
[tex]\[
2x + 1
\][/tex]
Therefore, the other linear factor of the trinomial [tex]\(2x^2 + 13x + 6\)[/tex], given that one factor is [tex]\(x + 6\)[/tex], is:
[tex]\[
\boxed{2x + 1}
\][/tex]
