Use synthetic division to find the quotient and the remainder of:

(x^3-7x^2+13x+3)/(x-2)

I will try to line up the numbers,

2 │ 1 -7 12 3
...------------
.......2 -10 6
....1 -5 ..3 9

so (x^3-7x^2+13x+3)/(x-2)
= x^2-5x+3 + 9/(x-2)

Do we add the 9 and 3 to get 12??

=x^2 -5x (12/x-2)

To find the quotient and remainder using synthetic division, we will divide (x^3 - 7x^2 + 13x + 3) by (x - 2).

Arrange the terms of the dividend in descending order:

1x^3 - 7x^2 + 13x + 3

Next, write the constant from the divisor (x - 2) on the left side:

2| 1 - 7 + 13 + 3

Divide the first term on the left side (1) by the divisor (2) and write the result above the line:

2| 1 - 7 + 13 + 3
-| 2
--------
1

Multiply the divisor (2) by the result (1) and write the product under the next term on the left side (-7):

2| 1 - 7 + 13 + 3
-| 2
--------
1 - 2

Subtract the product (-2) from the next term on the left side (-7) and write the result below the line:

2| 1 - 7 + 13 + 3
-| 2
--------
1 - 2
- 5

Repeat the process for the next terms:

2| 1 - 7 + 13 + 3
-| 2 - 5
--------
1 - 2 8

The quotient is: x^2 - 5x + 8
The remainder is: 8

Therefore, the quotient of (x^3 - 7x^2 + 13x + 3) divided by (x - 2) is x^2 - 5x + 8 with a remainder of 8.

To use synthetic division to find the quotient and remainder, follow these steps:

1. Write the polynomial in descending order of exponents (highest to lowest). In this case, the polynomial is: x^3 - 7x^2 + 13x + 3.

2. Identify the divisor. In this case, the divisor is x - 2.

3. Set up the synthetic division table:

2 | 1 -7 13 3

The numbers above the division bar represent the coefficients of the polynomial, starting with the highest degree term. If a term is missing, a zero coefficient is used as a placeholder.

4. Bring down the first coefficient, which is 1, next to the division bar:

2 | 1 -7 13 3
-----------
1

5. Multiply the divisor (2) by the number just brought down (1) and write the result under the next coefficient (-7):

2 | 1 -7 13 3
-----------
1
-14

6. Add the result to the next coefficient (13), and write the sum under the next coefficient:

2 | 1 -7 13 3
-----------
1
-14
-------
-1

7. Repeat the process until all coefficients are used:

2 | 1 -7 13 3
-----------
1
-14
-------
-1
4
-------
17

The numbers below the division bar represent the coefficients of the quotient, from left to right: 1, -1, and 17. The remainder is the last number in the third row, which is 17.

Therefore, the quotient is x^2 - x + 17, and the remainder is 17.