Use remainder theorem to factorize the following polynomial :
`2x^(3)+3x^(2)-9x-10`

Answer

To factorize the polynomial \( f(x) = 2x^3 + 3x^2 - 9x - 10 \) using the Remainder Theorem, we will follow these steps:

### Step 1: Identify the polynomial
Let \( f(x) = 2x^3 + 3x^2 - 9x - 10 \).

### Step 2: Use the Remainder Theorem
According to the Remainder Theorem, if \( f(c) = 0 \), then \( x - c \) is a factor of \( f(x) \). We will test some rational roots.

### Step 3: Test \( x = -1 \)
Calculate \( f(-1) \):
\[ f(-1) = 2(-1)^3 + 3(-1)^2 - 9(-1) - 10 \]
\[ = 2(-1) + 3(1) + 9 - 10 \]
\[ = -2 + 3 + 9 - 10 = 0 \]
Since \( f(-1) = 0 \), \( x + 1 \) is a factor of \( f(x) \).

### Step 4: Perform synthetic division
Now, we will divide \( f(x) \) by \( x + 1 \) using synthetic division.

Set up synthetic division with -1:
```
-1 | 2 3 -9 -10
| -2 -1 10
-----------------------
2 1 -10 0
```
The result is \( 2x^2 + x - 10 \).

### Step 5: Factor the quadratic
Now, we need to factor \( 2x^2 + x - 10 \). We will look for two numbers that multiply to \( 2 \times -10 = -20 \) and add to \( 1 \). The numbers are \( 5 \) and \( -4 \).

Rewrite the quadratic:
\[ 2x^2 + 5x - 4x - 10 = 0 \]
Group the terms:
\[ (2x^2 + 5x) + (-4x - 10) = 0 \]
Factor by grouping:
\[ x(2x + 5) - 2(2x + 5) = 0 \]
\[ (2x + 5)(x - 2) = 0 \]

### Step 6: Combine factors
Now we can write the complete factorization of \( f(x) \):
\[ f(x) = (x + 1)(2x + 5)(x - 2) \]

### Final Answer
Thus, the factorization of the polynomial \( 2x^3 + 3x^2 - 9x - 10 \) is:
\[ f(x) = (x + 1)(x - 2)(2x + 5) \]