Using the Remainder and Factor theorem, factorise the following polynomial :
`x^(3)+10x^(2)-37x+26`.

Answer

To factor the polynomial \( f(x) = x^3 + 10x^2 - 37x + 26 \) using the Remainder and Factor Theorem, we will follow these steps:

### Step 1: Identify Possible Rational Roots
We need to find the factors of the constant term (26) to identify possible rational roots. The factors of 26 are:
- \( \pm 1 \)
- \( \pm 2 \)
- \( \pm 13 \)
- \( \pm 26 \)

### Step 2: Test Possible Roots
We will substitute these values into the polynomial to see if any of them yield \( f(x) = 0 \).

1. **Testing \( x = 1 \)**:
\[ f(1) = 1^3 + 10(1^2) - 37(1) + 26 = 1 + 10 - 37 + 26 = 0 \]
Since \( f(1) = 0 \), \( x - 1 \) is a factor.

2. **Testing \( x = 2 \)**:
\[ f(2) = 2^3 + 10(2^2) - 37(2) + 26 = 8 + 40 - 74 + 26 = 0 \]
Since \( f(2) = 0 \), \( x - 2 \) is a factor.

3. **Testing \( x = -13 \)**:
\[ f(-13) = (-13)^3 + 10(-13)^2 - 37(-13) + 26 = -2197 + 1690 + 481 + 26 = 0 \]
Since \( f(-13) = 0 \), \( x + 13 \) is a factor.

### Step 3: Write the Factors
From our tests, we have found three factors:
- \( x - 1 \)
- \( x - 2 \)
- \( x + 13 \)

### Step 4: Write the Complete Factorization
The polynomial can be expressed as:
\[ f(x) = (x - 1)(x - 2)(x + 13) \]

### Conclusion
Thus, the factorization of the polynomial \( x^3 + 10x^2 - 37x + 26 \) is:
\[ f(x) = (x - 1)(x - 2)(x + 13) \]