To divide the polynomial [tex]\(3x^5 - 4x^3 + 3x^2 - 4x + 2\)[/tex] by [tex]\(x - 2\)[/tex] using synthetic division, follow these steps:
### Step 1: Set Up Synthetic Division
1. Identify the coefficients of the polynomial [tex]\(3x^5 - 4x^3 + 3x^2 - 4x + 2\)[/tex]:
- These coefficients are: [tex]\(3, 0, -4, 3, -4, 2\)[/tex].
- Note: We include a coefficient of 0 for the missing term [tex]\(x^4\)[/tex].
2. Identify the root of the divisor [tex]\(x - 2\)[/tex]:
- The root is [tex]\(2\)[/tex].
### Step 2: Synthetic Division Process
1. Write down the coefficients: [tex]\(3, 0, -4, 3, -4, 2\)[/tex].
2. Bring down the leading coefficient (first number): [tex]\(3\)[/tex].
Here is a structured table for the synthetic division:
```
2 | 3 0 -4 3 -4 2
| 6 12 16 38 68
-------------------------
3 6 8 19 34 70
```
### Step 3: Perform the Steps
1. Multiply the root [tex]\(2\)[/tex] by the brought down number [tex]\(3\)[/tex] and write it under the next coefficient:
- [tex]\(2 \times 3 = 6\)[/tex].
2. Add this result to the next coefficient (#2):
- [tex]\(0 + 6 = 6\)[/tex].
3. Continue this pattern:
- [tex]\(2 \times 6 = 12\)[/tex], and [tex]\( -4 + 12 = 8\)[/tex].
- [tex]\(2 \times 8 = 16\)[/tex], and [tex]\(3 + 16 = 19\)[/tex].
- [tex]\(2 \times 19 = 38\)[/tex], and [tex]\(-4 + 38 = 34\)[/tex].
- [tex]\(2 \times 34 = 68\)[/tex], and [tex]\(2 + 68 = 70\)[/tex].
### Step 4: Interpret the Result
- The last number 70 is the remainder.
- The rest of the numbers represent the coefficients of the quotient polynomial.
### Step 5: Write the Quotient Polynomial
- The quotient is: [tex]\(3x^4 + 6x^3 + 8x^2 + 19x + 34\)[/tex].
- The remainder is: [tex]\(70\)[/tex].
### Final Answer
[tex]\[
\frac{3x^5 - 4x^3 + 3x^2 - 4x + 2}{x - 2} = 3x^4 + 6x^3 + 8x^2 + 19x + 34 + \frac{70}{x - 2}
\][/tex]
