Sure, let's factor the polynomial [tex]\(x^2 - 12x + 27\)[/tex] step-by-step.
1. Understand the Polynomial: We start with the quadratic polynomial [tex]\(x^2 - 12x + 27\)[/tex].
2. Identify the Coefficients: The polynomial is in the standard form [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -12\)[/tex], and [tex]\(c = 27\)[/tex].
3. Find the Factors of the Constant Term: We need to find two numbers that multiply to [tex]\(27\)[/tex] and add up to [tex]\(-12\)[/tex].
- List the pairs of factors of [tex]\(27\)[/tex]: [tex]\((1, 27)\)[/tex], [tex]\((3, 9)\)[/tex], [tex]\((-3, -9)\)[/tex], etc.
4. Select the Correct Pair: From the pairs above, [tex]\((-3, -9)\)[/tex] multiply to [tex]\(27\)[/tex] and add up to [tex]\(-12\)[/tex].
5. Write the Factored Form:
- Using the numbers [tex]\(-3\)[/tex] and [tex]\(-9\)[/tex], we can write the factors of the polynomial as:
[tex]\[
(x - 3)(x - 9)
\][/tex]
Thus, the factored form of the polynomial [tex]\(x^2 - 12x + 27\)[/tex] is [tex]\((x - 9)(x - 3)\)[/tex].
So, the correct answer is:
[tex]\[
(x - 9)(x - 3)
\][/tex]
