Using Remainder Theorem, factorise : `x^(3)+10x^2-37x+26` completely.

Answer

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

### Step 1: Identify the polynomial
Let \( f(x) = x^3 + 10x^2 - 37x + 26 \).

### Step 2: Check for rational roots using the Remainder Theorem
We will check for possible rational roots by evaluating \( f(1) \):
\[ f(1) = 1^3 + 10(1^2) - 37(1) + 26 = 1 + 10 - 37 + 26 = 0 \]
Since \( f(1) = 0 \), by the Remainder Theorem, \( x - 1 \) is a factor of \( f(x) \).

### Step 3: Perform synthetic division
Now we will divide \( f(x) \) by \( x - 1 \) using synthetic division.

1. Write the coefficients: \( 1, 10, -37, 26 \).
2. Set up synthetic division with \( 1 \):
- Bring down the \( 1 \).
- Multiply \( 1 \) by \( 1 \) (the root) and add to the next coefficient:
- \( 10 + 1 = 11 \)
- Multiply \( 1 \) by \( 11 \) and add to the next coefficient:
- \( -37 + 11 = -26 \)
- Multiply \( 1 \) by \( -26 \) and add to the next coefficient:
- \( 26 - 26 = 0 \)

The result of the synthetic division is:
\[ x^2 + 11x - 26 \]

### Step 4: Factor the quadratic
Now, we need to factor \( x^2 + 11x - 26 \). We look for two numbers that multiply to \(-26\) and add to \(11\). The numbers \(13\) and \(-2\) fit this requirement.

Thus, we can write:
\[ x^2 + 11x - 26 = (x + 13)(x - 2) \]

### Step 5: Write the complete factorization
Now we can write the complete factorization of \( f(x) \):
\[ f(x) = (x - 1)(x + 13)(x - 2) \]

### Final Answer
The complete factorization of \( x^3 + 10x^2 - 37x + 26 \) is:
\[ f(x) = (x - 1)(x + 13)(x - 2) \]

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