To determine the correct factorization of the polynomial [tex]\( 2x^2 - 12x + 18 \)[/tex], we should follow a systematic approach to identify the correct factorization form. Here's a step-by-step solution:
1. Identify the polynomial: The polynomial given is [tex]\( 2x^2 - 12x + 18 \)[/tex].
2. Check for common factors:
Notice that each term in the polynomial [tex]\( 2x^2 - 12x + 18 \)[/tex] can be divided by 2. Factor out the common factor:
[tex]\[
2x^2 - 12x + 18 = 2(x^2 - 6x + 9)
\][/tex]
3. Factor the quadratic expression inside the parentheses:
Focus on the quadratic expression [tex]\( x^2 - 6x + 9 \)[/tex]:
- Find two numbers that multiply to [tex]\( 9 \)[/tex] (the constant term) and add up to [tex]\(-6\)[/tex] (the coefficient of the linear term).
- The numbers that satisfy this are [tex]\(-3\)[/tex] and [tex]\(-3\)[/tex] (since [tex]\(-3 \times -3 = 9\)[/tex] and [tex]\(-3 + (-3) = -6\)[/tex]).
Therefore, the quadratic expression can be factored as:
[tex]\[
x^2 - 6x + 9 = (x - 3)(x - 3) = (x - 3)^2
\][/tex]
4. Substitute back into the original factorization:
Now substitute back into the factored expression:
[tex]\[
2(x^2 - 6x + 9) = 2((x - 3)^2) = 2(x - 3)^2
\][/tex]
So, the correct factorization of the polynomial [tex]\( 2x^2 - 12x + 18 \)[/tex] is:
[tex]\[
2(x - 3)^2
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{2(x - 3)^2}
\][/tex]
This corresponds to option A in the given choices. Therefore, the correct answer is A. [tex]\(2(x - 3)^2\)[/tex].
