Let's factorize the given trinomial:
[tex]\[
2x^2 - 16x + 30
\][/tex]
To factorize this trinomial, we look for two binomials multiplied together that give us the original polynomial when expanded. Here's the step-by-step process:
1. Identify the quadratic equation:
The given polynomial is in the form [tex]\(ax^2 + bx + c\)[/tex].
Here, [tex]\(a = 2\)[/tex], [tex]\(b = -16\)[/tex], and [tex]\(c = 30\)[/tex].
2. Find the factors of [tex]\(ac\)[/tex]:
We need to find two numbers that multiply to [tex]\(a \times c\)[/tex] (i.e., [tex]\(2 \times 30 = 60\)[/tex]) and add up to [tex]\(b\)[/tex] (i.e., [tex]\(-16\)[/tex]).
3. Determine the pairs of factors of 60:
The factors of 60 are:
[tex]\[
(1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10)
\][/tex]
4. Identify the correct pair that adds to -16:
We need a pair of factors whose product is 60 and whose sum is -16. The correct pair is [tex]\(-6\)[/tex] and [tex]\(-10\)[/tex]:
[tex]\[
-6 \times -10 = 60 \quad \text{and} \quad -6 + (-10) = -16
\][/tex]
5. Rewrite the middle term using these factors:
We can decompose [tex]\(-16x\)[/tex] into [tex]\(-6x - 10x\)[/tex]:
[tex]\[
2x^2 - 6x - 10x + 30
\][/tex]
6. Factor by grouping:
Group the terms in pairs and factor out the common factors:
[tex]\[
(2x^2 - 6x) + (-10x + 30)
\][/tex]
Factor out the common factors in each group:
[tex]\[
2x(x - 3) - 10(x - 3)
\][/tex]
7. Factor out the common binomial factor [tex]\((x - 3)\)[/tex]:
[tex]\[
(2x - 10)(x - 3)
\][/tex]
8. Simplify the expression [tex]\((2x - 10)\)[/tex]:
We can factor out a 2 from [tex]\((2x - 10)\)[/tex]:
[tex]\[
2(x - 5)(x - 3)
\][/tex]
Therefore, the factorization of the trinomial [tex]\(2x^2 - 16x + 30\)[/tex] is:
[tex]\[
2(x - 5)(x - 3)
\][/tex]
Comparing with the given options, the correct answer is:
A. [tex]\(2(x-3)(x-5)\)[/tex]
However, note that the order of factors (x-5) and (x-3) does not affect the product, so the correct final answer aligning with the options is:
A. [tex]\(2(x-3)(x-5)\)[/tex]
