To find the factors of the polynomial [tex]\(g^2 - 32g + 256\)[/tex], we can use the method of factoring quadratic equations. Here's the step-by-step solution:
1. Identify the coefficient (a, b, c) in the quadratic equation:
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c\)[/tex].
In this case:
- [tex]\(a = 1\)[/tex] (coefficient of [tex]\(g^2\)[/tex])
- [tex]\(b = -32\)[/tex] (coefficient of [tex]\(g\)[/tex])
- [tex]\(c = 256\)[/tex] (constant term)
2. Find two numbers that multiply to [tex]\(a \cdot c\)[/tex] (which is [tex]\(1 \cdot 256 = 256\)[/tex]) and add up to [tex]\(b\)[/tex] (which is [tex]\(-32\)[/tex]):
We need to find two numbers whose product is 256 and whose sum is -32.
3. List the pairs of factors of 256:
The pairs of factors of 256 are:
- [tex]\(1 \cdot 256\)[/tex]
- [tex]\(2 \cdot 128\)[/tex]
- [tex]\(4 \cdot 64\)[/tex]
- [tex]\(8 \cdot 32\)[/tex]
- [tex]\(16 \cdot 16\)[/tex]
4. Determine which pair of factors adds up to -32:
- The correct pair is [tex]\(-16\)[/tex] and [tex]\(-16\)[/tex] because [tex]\((-16) \cdot (-16) = 256\)[/tex] and [tex]\(-16 + (-16) = -32\)[/tex].
5. Express the middle term using the found pair:
Rewrite [tex]\(-32g\)[/tex] as [tex]\(-16g - 16g\)[/tex]:
[tex]\[
g^2 - 32g + 256 = g^2 - 16g - 16g + 256
\][/tex]
6. Group the terms and factor by grouping:
Group the terms in pairs:
[tex]\[
(g^2 - 16g) - (16g - 256)
\][/tex]
7. Factor out the common factors in each group:
[tex]\[
g(g - 16) - 16(g - 16)
\][/tex]
8. Factor out the common binomial factor [tex]\((g - 16)\)[/tex]:
[tex]\[
(g - 16)(g - 16)
\][/tex]
9. Write down the factors:
The factors of [tex]\(g^2 - 32g + 256\)[/tex] are [tex]\((g - 16)\)[/tex] and [tex]\((g - 16)\)[/tex], which can just be written as [tex]\((g - 16)^2\)[/tex].
Thus, one of the factors of [tex]\(g^2 - 32g + 256\)[/tex] is [tex]\(\boxed{g - 16}\)[/tex].
