using synthetic division step by step.
3x^2-2x+7 by x-2
Synthetic division is a shorthand method of dividing a polynomial by a binomial of the form x - c, where c is a constant. Here, we want to divide the polynomial 3x^2 - 2x + 7 by x - 2. The constant c in this case is 2.
Here's the step-by-step process for synthetic division:
Step 1: Write down the coefficients of the polynomial you want to divide. For the polynomial 3x^2 - 2x + 7, the coefficients are 3, -2, and 7.
Step 2: Write down the value of c from the binomial x - c that the polynomial is being divided by. Since we're dividing by x - 2, c is 2.
Step 3: Set up the synthetic division framework. You write the value of c on the left side and the coefficients on the right side, separated by a vertical line:
```
__________
2 | 3 -2 7
```
Step 4: Bring down the leading coefficient. This is your new first coefficient:
```
3
__________
2 | 3 -2 7
```
Step 5: Multiply the value that you just brought down (the leading coefficient) by c, and write the result under the next coefficient:
```
3
__________
2 | 3 -2 7
(3*2)
```
```
3
__________
2 | 3 -2 7
6
```
Step 6: Add the value you wrote down to the next coefficient and write the result below:
```
3
__________
2 | 3 -2 7
6
-------
4
```
Step 7: Repeat the same process for the next column. Multiply the result you just got by c, and write the result under the next coefficient:
```
3 4
__________
2 | 3 -2 7
6 8
-------
4 15
```
Step 8: The numbers at the bottom are the coefficients of the quotient. In our case, the result of the synthetic division is the polynomial 3x + 4 with a remainder of 15.
The final answer in polynomial form is:
3x + 4 + 15/(x - 2)
This result tells you that the original polynomial 3x^2 - 2x + 7 divided by x - 2 equals 3x + 4 with a remainder of 15.