To solve the given polynomial division [tex]\((3x^4 + 6x^3 + 2x^2 + 9x + 10) \div (x + 2)\)[/tex] using synthetic division, we follow these steps:
1. Identify the divisor:
The polynomial divisor is [tex]\(x + 2\)[/tex], which corresponds to a value of [tex]\(x = -2\)[/tex].
2. List the coefficients of the polynomial:
The coefficients of the polynomial [tex]\(3x^4 + 6x^3 + 2x^2 + 9x + 10\)[/tex] are:
[tex]\[ [3, 6, 2, 9, 10] \][/tex]
3. Set up the synthetic division table:
We start by writing down the coefficients in a row and place the zero of the divisor (which is -2) to the left.
4. Performing synthetic division:
- Bring down the first coefficient (3) as it is.
- Multiply the value brought down by -2 and write the product under the next coefficient.
- Add the product to the next coefficient and write the sum below it.
- Continue this process until all coefficients have been used.
Here's the detailed step-by-step process:
| Coefficients | 3 | 6 | 2 | 9 | 10 |
|----------------------|----|----|----|----|-----|
| | | -6 | -12| -26| -52 |
| Bring down the first | 3 | 0 | 2 | 5 | 0 |
- Bring down [tex]\(3\)[/tex].
- Multiply [tex]\(3\)[/tex] by [tex]\(-2\)[/tex] to get [tex]\(-6\)[/tex]; add [tex]\(-6\)[/tex] to [tex]\(6\)[/tex] to get [tex]\(0\)[/tex].
- Multiply [tex]\(0\)[/tex] by [tex]\(-2\)[/tex] to get [tex]\(0\)[/tex]; add [tex]\(0\)[/tex] to [tex]\(2\)[/tex] to get [tex]\(2\)[/tex].
- Multiply [tex]\(2\)[/tex] by [tex]\(-2\)[/tex] to get [tex]\(-4\)[/tex]; add [tex]\(-4\)[/tex] to [tex]\(9\)[/tex] to get [tex]\(5\)[/tex].
- Multiply [tex]\(5\)[/tex] by [tex]\(-2\)[/tex] to get [tex]\(-10\)[/tex]; add [tex]\(-10\)[/tex] to [tex]\(10\)[/tex] to get [tex]\(0\)[/tex].
5. Result:
The row obtained from the above steps gives us the coefficients of the quotient and the remainder term:
- Quotient coefficients: [tex]\([3, 0, 2, 5]\)[/tex]
- Remainder: [tex]\(0\)[/tex]
Thus, the quotient polynomial is [tex]\(3x^3 + 0x^2 + 2x + 5\)[/tex], which simplifies to:
[tex]\[ \boxed{3x^3 + 2x + 5} \][/tex]
Hence, the correct representation of the quotient is [tex]\(3 x^3 + 2 x + 5\)[/tex].
