Divide the polynomical `6x^(4)-44x^(2)+6x-3` by the polynomial `x^(2)-3x+` 1 and verify the division algorithm.

Answer

To divide the polynomial \(6x^4 - 44x^2 + 6x - 3\) by the polynomial \(x^2 - 3x + 1\), we will follow the polynomial long division method. Let's break it down step by step.

### Step 1: Set Up the Division
We have:
- Dividend (the polynomial to be divided): \(6x^4 - 44x^2 + 6x - 3\)
- Divisor (the polynomial we are dividing by): \(x^2 - 3x + 1\)

### Step 2: Divide the Leading Terms
Divide the leading term of the dividend by the leading term of the divisor:
\[ \frac{6x^4}{x^2} = 6x^2 \]
This gives us the first term of the quotient.

### Step 3: Multiply and Subtract
Now, multiply the entire divisor by \(6x^2\):
\[ 6x^2 \cdot (x^2 - 3x + 1) = 6x^4 - 18x^3 + 6x^2 \]
Now, subtract this from the original polynomial:
\[ (6x^4 - 44x^2 + 6x - 3) - (6x^4 - 18x^3 + 6x^2) \]
Distributing the negative sign:
\[ 6x^4 - 44x^2 + 6x - 3 - 6x^4 + 18x^3 - 6x^2 \]
This simplifies to:
\[ 18x^3 - 38x^2 + 6x - 3 \]

### Step 4: Repeat the Process
Now, we will repeat the process with the new polynomial \(18x^3 - 38x^2 + 6x - 3\).

1. Divide the leading term:
\[ \frac{18x^3}{x^2} = 18x \]
2. Multiply the divisor by \(18x\):
\[ 18x \cdot (x^2 - 3x + 1) = 18x^3 - 54x^2 + 18x \]
3. Subtract:
\[ (18x^3 - 38x^2 + 6x - 3) - (18x^3 - 54x^2 + 18x) \]
This simplifies to:
\[ 16x^2 - 12x - 3 \]

### Step 5: Continue the Division
Now, we will divide \(16x^2 - 12x - 3\) by \(x^2 - 3x + 1\).

1. Divide the leading term:
\[ \frac{16x^2}{x^2} = 16 \]
2. Multiply the divisor by \(16\):
\[ 16 \cdot (x^2 - 3x + 1) = 16x^2 - 48x + 16 \]
3. Subtract:
\[ (16x^2 - 12x - 3) - (16x^2 - 48x + 16) \]
This simplifies to:
\[ 36x - 19 \]

### Step 6: Final Remainder
Now, we have a polynomial \(36x - 19\) which has a degree less than the divisor \(x^2 - 3x + 1\). Thus, we stop here.

### Final Result
The quotient is:
\[ 6x^2 + 18x + 16 \]
And the remainder is:
\[ 36x - 19 \]

### Step 7: Verify the Division Algorithm
According to the division algorithm:
\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]
Substituting the values:
\[ 6x^4 - 44x^2 + 6x - 3 = (x^2 - 3x + 1)(6x^2 + 18x + 16) + (36x - 19) \]
We can expand the right-hand side and simplify to check if it equals the left-hand side.