Factories `9x^(3)-27x^(2)-100x+300`

Answer

To factor the polynomial \( 9x^3 - 27x^2 - 100x + 300 \), we will follow these steps:

### Step 1: Identify the Polynomial
Let \( p(x) = 9x^3 - 27x^2 - 100x + 300 \).

### Step 2: Use the Rational Root Theorem
We will look for rational roots (factors of the constant term) to find one factor of the polynomial. The constant term is 300. The factors of 300 include:
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300.

### Step 3: Test Possible Rational Roots
Let's test \( x = 3 \):
\[ p(3) = 9(3)^3 - 27(3)^2 - 100(3) + 300 \]
Calculating each term:
- \( 9(3^3) = 9 \times 27 = 243 \)
- \( -27(3^2) = -27 \times 9 = -243 \)
- \( -100(3) = -300 \)
- \( +300 = 300 \)

Now, substituting these values into \( p(3) \):
\[ p(3) = 243 - 243 - 300 + 300 = 0 \]
Since \( p(3) = 0 \), \( x - 3 \) is a factor of \( p(x) \).

### Step 4: Perform Polynomial Long Division
Now, we will divide \( p(x) \) by \( x - 3 \).

1. Divide the leading term: \( \frac{9x^3}{x} = 9x^2 \).
2. Multiply \( 9x^2 \) by \( x - 3 \):
\[ 9x^2(x - 3) = 9x^3 - 27x^2 \]
3. Subtract this from \( p(x) \):
\[ (9x^3 - 27x^2 - 100x + 300) - (9x^3 - 27x^2) = -100x + 300 \]
4. Bring down the next term:
\[ -100x + 300 \]
5. Divide the leading term: \( \frac{-100x}{x} = -100 \).
6. Multiply \( -100 \) by \( x - 3 \):
\[ -100(x - 3) = -100x + 300 \]
7. Subtract:
\[ (-100x + 300) - (-100x + 300) = 0 \]

The result of the division is \( 9x^2 - 100 \).

### Step 5: Factor the Quadratic
Now we need to factor \( 9x^2 - 100 \). This is a difference of squares:
\[ 9x^2 - 100 = (3x)^2 - (10)^2 = (3x - 10)(3x + 10) \]

### Step 6: Write the Final Factorization
Combining all the factors, we have:
\[ p(x) = (x - 3)(3x - 10)(3x + 10) \]

### Final Answer
The complete factorization of the polynomial \( 9x^3 - 27x^2 - 100x + 300 \) is:
\[ (x - 3)(3x - 10)(3x + 10) \]

---