Synthetic division is a shorthand method of dividing a polynomial by a linear factor. In this case, we are looking to divide the polynomial x³ + 1x + 1 by a factor that we’ll determine as we start the process.
First, we need to express our polynomial in standard form, which gives us:
- x³ + 0x² + 1x + 1
Now, we need to find a root (or a number ‘c’) that will help us in synthetic division. Looking at the coefficients of our polynomial, we can factor out variables around simple roots like 1. A good candidate for c is -1 because it’s simple and intuitive.
We set up synthetic division using -1:
- Write the coefficients: 1 (for x³), 0 (for x²), 1 (for x), and 1 (for the constant term).
Now, set up the synthetic division:
- -1 | 1 0 1 1
Now, bring down the leading coefficient (1):
- -1 | 1 0 1 1
- | 1
- | 1
Next, multiply -1 by 1 and add to the next coefficient (0):
- -1 | 1 0 1 1
- | -1
- | 1
Now, take 1 (the result) and do the same with the next coefficient (1):
- -1 | 1 0 1 1
- | -1 1
- | 1 0
Finally, multiply -1 by 0 and add to the final coefficient (1):
- -1 | 1 0 1 1
- | -1 1 0
- | 1 0
Now you should have the last result, which is 0, and the bottom row gives you the coefficients of the quotient:
- 1 (for x²)
- 1 (for x)
- 1 (constant)
This means the quotient of the polynomial x³ + 1x + 1 divided by (x + 1) is:
x² + 1x + 1
Thus, the final quotient is:
Q = x² + x + 1