Sure, let's perform the polynomial division step-by-step to divide [tex]\(3x^3 - 5x^2 + x - 2\)[/tex] by [tex]\(x - 2\)[/tex].
### Step 1: Setup the division
We start by setting up the polynomial division with [tex]\(3x^3 - 5x^2 + x - 2\)[/tex] as the dividend and [tex]\(x - 2\)[/tex] as the divisor.
```
Dividend: 3x^3 - 5x^2 + x - 2
Divisor: x - 2
```
### Step 2: Divide the leading terms
First, divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{3x^3}{x} = 3x^2
\][/tex]
### Step 3: Multiply and subtract
Next, multiply the entire divisor by [tex]\(3x^2\)[/tex] and subtract this product from the dividend:
1. Multiply:
[tex]\[
(x - 2) \times 3x^2 = 3x^3 - 6x^2
\][/tex]
2. Subtract:
[tex]\[
(3x^3 - 5x^2 + x - 2) - (3x^3 - 6x^2) = (3x^3 - 3x^3) + (-5x^2 + 6x^2) + x - 2 = x^2 + x - 2
\][/tex]
### Step 4: Repeat the process
Continue the same process with the new polynomial [tex]\(x^2 + x - 2\)[/tex]:
1. Divide the leading terms:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
2. Multiply:
[tex]\[
(x - 2) \times x = x^2 - 2x
\][/tex]
3. Subtract:
[tex]\[
(x^2 + x - 2) - (x^2 - 2x) = (x^2 - x^2) + (x + 2x) - 2 = 3x - 2
\][/tex]
### Step 5: Repeat the process again
Continue with the new polynomial [tex]\(3x - 2\)[/tex]:
1. Divide the leading terms:
[tex]\[
\frac{3x}{x} = 3
\][/tex]
2. Multiply:
[tex]\[
(x - 2) \times 3 = 3x - 6
\][/tex]
3. Subtract:
[tex]\[
(3x - 2) - (3x - 6) = (3x - 3x) + (-2 + 6) = 4
\][/tex]
### Step 6: Compile the quotient and remainder
After completing the division process, we find that:
- Quotient: [tex]\(3x^2 + x + 3\)[/tex]
- Remainder: [tex]\(4\)[/tex]
Therefore, the result of the division [tex]\((3x^3 - 5x^2 + x - 2) \div (x - 2)\)[/tex] is a quotient of [tex]\(3x^2 + x + 3\)[/tex] with a remainder of [tex]\(4\)[/tex].
