divide using synthetic division
(x^3-3x^2-10) ÷ (x+3)
Synthetic division is used to divide a polynomial by a binomial of the form (x - c). In this case, the divisor is (x + 3) which is the same as (x - (-3)). The "c" here is -3.
To perform synthetic division, we'll use -3 in the process. Here's the setup for the polynomial \( x^3 - 3x^2 + 0x - 10 \) (notice that we included a "0x" term for the missing x-term):
Step 1: Write down the coefficients of the polynomial and the constant term. If any terms are missing, use 0 as the coefficient for that term.
\( x^3 \) has a coefficient of 1.
\( x^2 \) has a coefficient of -3.
\( x^1 \) (which is not present) will have a coefficient of 0.
\( x^0 \) (the constant term) is -10.
Step 2: Set up the synthetic division.
```
-3 | 1 -3 0 -10
|-----------------
|
```
Step 3: Bring down the leading coefficient.
```
-3 | 1 -3 0 -10
|-----------------
| 1
```
Step 4: Multiply -3 by the number brought down (1) and write the result under the next coefficient.
```
-3 | 1 -3 0 -10
| -3
|-----------------
| 1
```
Step 5: Add the numbers in the second column to get the new coefficient, and then repeat the process of multiplying by -3 and adding, working your way across.
```
-3 | 1 -3 0 -10
| -3 9 -27
|-----------------
| 1 -6 9 -37
```
The numbers on the bottom row are the coefficients of the resulting polynomial, starting from one degree less than the original (since we "lost" an x by dividing):
The quotient with the remainder is: \( x^2 - 6x + 9 - \frac{37}{x+3} \).
However, in synthetic division, we often leave off the remainder part and say the result of \( x^3 - 3x^2 - 10 \) divided by \( x + 3 \) is \( x^2 - 6x + 9 \) with a remainder of -37.