To understand which factored form corresponds to the polynomial [tex]\(x^2 + 6x - 27\)[/tex], let's start by breaking down the polynomial and examining each option.
### Step-by-Step Solution:
1. Given Polynomial:
[tex]\[ x^2 + 6x - 27 \][/tex]
2. Factoring [tex]\(x^2 + 6x - 27\)[/tex]:
To factor this, let's explore each proposed factored form. For each option, we will expand the factored form and compare it with the given polynomial [tex]\(x^2 + 6x - 27\)[/tex].
3. Option (1):
[tex]\[ x^2 + 3x - 9x - 27 = (x+3)(x-9) \][/tex]
Expand:
[tex]\[ (x + 3)(x - 9) = x \cdot x + x \cdot (-9) + 3 \cdot x + 3 \cdot (-9) \][/tex]
[tex]\[ = x^2 - 9x + 3x - 27 \][/tex]
[tex]\[ = x^2 - 6x - 27 \][/tex]
This does not match the given polynomial [tex]\(x^2 + 6x - 27\)[/tex].
4. Option (2):
[tex]\[ x^2 + 6x - 3x - 27 = (x+6)(x-3) \][/tex]
Expand:
[tex]\[ (x + 6)(x - 3) = x \cdot x + x \cdot (-3) + 6 \cdot x + 6 \cdot (-3) \][/tex]
[tex]\[ = x^2 - 3x + 6x - 18 \][/tex]
[tex]\[ = x^2 + 3x - 18 \][/tex]
This does not match the given polynomial [tex]\(x^2 + 6x - 27\)[/tex].
5. Option (3):
[tex]\[ x^2 + 9x - 3x - 27 = (x+9)(x-3) \][/tex]
Expand:
[tex]\[ (x + 9)(x - 3) = x \cdot x + x \cdot (-3) + 9 \cdot x + 9 \cdot (-3) \][/tex]
[tex]\[ = x^2 - 3x + 9x - 27 \][/tex]
[tex]\[ = x^2 + 6x - 27 \][/tex]
This matches the given polynomial [tex]\(x^2 + 6x - 27\)[/tex].
6. Option (4):
[tex]\[ x^2 + 3x - 6x - 27 = (x+3)(x-6) \][/tex]
Expand:
[tex]\[ (x + 3)(x - 6) = x \cdot x + x \cdot (-6) + 3 \cdot x + 3 \cdot (-6) \][/tex]
[tex]\[ = x^2 - 6x + 3x - 18 \][/tex]
[tex]\[ = x^2 - 3x - 18 \][/tex]
This does not match the given polynomial [tex]\(x^2 + 6x - 27\)[/tex].
Thus, the correct factored form of [tex]\(x^2 + 6x - 27\)[/tex] is shown in Option (3):
[tex]\[ x^2 + 9x - 3x - 27 = (x+9)(x-3) \][/tex]
