To factorize the polynomial [tex]\( P(x) = x^4 - 4x^3 + 4x^2 - 36x - 45 \)[/tex] given that the roots are 5 and -1, we will follow a systematic step-by-step process:
### Step 1: Verify that 5 and -1 are roots
We know from the problem statement that 5 and -1 are roots. Therefore, [tex]\(x - 5\)[/tex] and [tex]\(x + 1\)[/tex] are factors of the polynomial.
### Step 2: Divide the polynomial by [tex]\((x - 5)\)[/tex] and [tex]\((x + 1)\)[/tex]
First, we will divide [tex]\( P(x) \)[/tex] by [tex]\((x - 5)\)[/tex]:
[tex]\[ P(x) = (x - 5)Q(x) \][/tex]
After division, [tex]\( Q(x) \)[/tex] represents the quotient.
Next, we will divide the obtained quotient [tex]\( Q(x) \)[/tex] by [tex]\((x + 1)\)[/tex]:
[tex]\[ Q(x) = (x + 1)R(x) \][/tex]
After this division, [tex]\( R(x) \)[/tex] represents the remaining quotient.
### Step 3: Find the factorization of the remaining quotient [tex]\( R(x) \)[/tex]
The quotient [tex]\( R(x) \)[/tex] will be a quadratic polynomial. We then need to factorize [tex]\( R(x) \)[/tex] completely to find its roots.
### Step 4: Combine all factors
Combining the factors obtained from [tex]\(5\)[/tex] and [tex]\(-1\)[/tex] with the factorized form of [tex]\( R(x) \)[/tex], we will get the complete factorization of [tex]\( P(x) \)[/tex].
### Result
Carrying out the steps described, we arrive at the complete factorization:
[tex]\[ P(x) = (x - 5)(x + 1)(x^2 + 9) \][/tex]
Given the options:
a) [tex]\((x + 5)(x - 1)(x - 3i)(x + 3i)\)[/tex]
b) [tex]\((x - 5)(x + 1)(x - 3i)(x + 3i)\)[/tex]
c) [tex]\((n - 5)(n \perp 1)(n^2 - 0)\)[/tex]
The correct answer is:
b) [tex]\((x - 5)(x + 1)(x - 3i)(x + 3i)\)[/tex]
Breaking it down:
- [tex]\((x - 5)\)[/tex] and [tex]\((x + 1)\)[/tex] are given roots corresponding to 5 and -1.
- The quadratic [tex]\(x^2 + 9\)[/tex] can be factored further over the complex numbers to [tex]\((x - 3i)(x + 3i)\)[/tex].
Therefore, the complete factorization of [tex]\( P(x) \)[/tex] is:
[tex]\((x - 5)(x + 1)(x - 3i)(x + 3i)\)[/tex].
