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.