To determine which expression is equivalent to the given polynomial [tex]\(2x^2 - 11x - 6\)[/tex], we need to factor the polynomial. A step-by-step approach to factorizing a quadratic polynomial [tex]\(ax^2 + bx + c\)[/tex] is as follows:
1. Identify the coefficients: For the given polynomial [tex]\(2x^2 - 11x - 6\)[/tex], we have:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -11\)[/tex]
- [tex]\(c = -6\)[/tex]
2. Find two numbers that multiply to [tex]\(a \cdot c\)[/tex] (the product of the coefficients of [tex]\(x^2\)[/tex] and the constant term) and add up to [tex]\(b\)[/tex] (the coefficient of [tex]\(x\)[/tex]):
- [tex]\(a \cdot c = 2 \cdot (-6) = -12\)[/tex]
- We need two numbers that multiply to [tex]\(-12\)[/tex] and add up to [tex]\(-11\)[/tex]:
- These numbers are [tex]\(-12\)[/tex] and [tex]\(1\)[/tex] because [tex]\(-12 \cdot 1 = -12\)[/tex] and [tex]\(-12 + 1 = -11\)[/tex].
3. Rewrite the middle term (-11x) using the two numbers found:
[tex]\[
2x^2 - 11x - 6 = 2x^2 - 12x + x - 6
\][/tex]
4. Factor by grouping:
- Group the terms into two pairs:
[tex]\[
(2x^2 - 12x) + (x - 6)
\][/tex]
- Factor out the greatest common factor from each pair:
[tex]\[
2x(x - 6) + 1(x - 6)
\][/tex]
- Factor out the common binomial factor [tex]\((x - 6)\)[/tex]:
[tex]\[
(2x + 1)(x - 6)
\][/tex]
Thus, the polynomial [tex]\(2x^2 - 11x - 6\)[/tex] factors to [tex]\((2x + 1)(x - 6)\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{(2x+1)(x-6)}
\][/tex]
Checking the choices given:
A. [tex]\((2x + 3)(x - 2)\)[/tex]
B. [tex]\(2(x - 3)(x + 1)\)[/tex]
C. [tex]\((2x + 1)(x - 6)\)[/tex]
D. [tex]\(2(x + 3)(x - 2)\)[/tex]
The correct choice is [tex]\(C\)[/tex]: [tex]\((2x + 1)(x - 6)\)[/tex].
