Divide `x^(4)+5x^(3)-x^(2)+4x-3` by x-2 and verify the division algorithm.

Explanation

To divide the polynomial \( p(x) = x^4 + 5x^3 - x^2 + 4x - 3 \) by \( g(x) = x - 2 \) and verify the division algorithm, we will follow the polynomial long division method step by step.

### Step 1: Set Up the Division
We start with the polynomial \( p(x) = x^4 + 5x^3 - x^2 + 4x - 3 \) and we want to divide it by \( g(x) = x - 2 \).

### Step 2: Divide the Leading Terms
To begin the division, we divide the leading term of \( p(x) \) by the leading term of \( g(x) \):
\[ \frac{x^4}{x} = x^3 \]
So, the first term of our quotient \( q(x) \) is \( x^3 \).

### Step 3: Multiply and Subtract
Next, we multiply \( x^3 \) by \( g(x) \):
\[ x^3 \cdot (x - 2) = x^4 - 2x^3 \]
Now, we subtract this from \( p(x) \):
\[ (x^4 + 5x^3 - x^2 + 4x - 3) - (x^4 - 2x^3) = (5x^3 + 2x^3) - x^2 + 4x - 3 = 7x^3 - x^2 + 4x - 3 \]

### Step 4: Repeat the Process
Now we repeat the process with the new polynomial \( 7x^3 - x^2 + 4x - 3 \).

1. Divide the leading term:
\[ \frac{7x^3}{x} = 7x^2 \]
So, the next term of \( q(x) \) is \( 7x^2 \).

2. Multiply and subtract:
\[ 7x^2 \cdot (x - 2) = 7x^3 - 14x^2 \]
Now, subtract:
\[ (7x^3 - x^2 + 4x - 3) - (7x^3 - 14x^2) = (14x^2 - x^2) + 4x - 3 = 13x^2 + 4x - 3 \]

### Step 5: Continue the Division
1. Divide the leading term:
\[ \frac{13x^2}{x} = 13x \]
Add \( 13x \) to \( q(x) \).

2. Multiply and subtract:
\[ 13x \cdot (x - 2) = 13x^2 - 26x \]
Now, subtract:
\[ (13x^2 + 4x - 3) - (13x^2 - 26x) = (4x + 26x) - 3 = 30x - 3 \]

### Step 6: Final Division
1. Divide the leading term:
\[ \frac{30x}{x} = 30 \]
Add \( 30 \) to \( q(x) \).

2. Multiply and subtract:
\[ 30 \cdot (x - 2) = 30x - 60 \]
Now, subtract:
\[ (30x - 3) - (30x - 60) = -3 + 60 = 57 \]

### Step 7: Write the Result
Now we have:
- Quotient: \( q(x) = x^3 + 7x^2 + 13x + 30 \)
- Remainder: \( r(x) = 57 \)

### Step 8: Verify the Division Algorithm
According to the division algorithm:
\[ p(x) = q(x) \cdot g(x) + r(x) \]
Substituting the values:
\[ p(x) = (x^3 + 7x^2 + 13x + 30)(x - 2) + 57 \]

Calculating the right-hand side:
1. Expand \( q(x) \cdot g(x) \):
\[ (x^3 + 7x^2 + 13x + 30)(x - 2) = x^4 + 7x^3 + 13x^2 + 30x - 2x^3 - 14x^2 - 26x - 60 \]
Combine like terms:
\[ x^4 + (7x^3 - 2x^3) + (13x^2 - 14x^2) + (30x - 26x) - 60 = x^4 + 5x^3 - x^2 + 4x - 60 \]

2. Add the remainder:
\[ x^4 + 5x^3 - x^2 + 4x - 60 + 57 = x^4 + 5x^3 - x^2 + 4x - 3 \]

This matches \( p(x) \), thus verifying the division algorithm.

### Final Result
- Quotient \( q(x) = x^3 + 7x^2 + 13x + 30 \)
- Remainder \( r(x) = 57 \)