Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form (x – c). In this case, you want to divide the polynomial x4 + x + 1 using synthetic division. To demonstrate this, let’s assume you want to divide it by (x – r), where r is a root of the polynomial. Since the polynomial doesn’t seem to have obvious rational roots, let’s first try common rational root testing or approximate values.
For synthetic division, pick an approximate root (for instance, let’s try r = 1). The coefficients of your polynomial are: 1 (for x^4), 0 (for x^3), 1 (for x^2), 0 (for x), and 1 (constant term). This can be represented as:
- 1 (coefficient of x4)
- 0 (coefficient of x3)
- 1 (coefficient of x2)
- 0 (coefficient of x)
- 1 (constant term)
Now, we’ll perform synthetic division:
1 | 1 0 1 0 1
| 1 1 2 2
-----------------------
1 1 2 2 3
Here’s how you read the synthetic division:
- Bring down the first coefficient (1).
- Multiply this value by r (1) and add it to the next coefficient (0). You get 1.
- Repeat this process for the remaining coefficients. The next value becomes 2, and so on.
The final row (1, 1, 2, 2) represents the coefficients of the quotient polynomial x3 + x2 + 2x + 2, with a remainder of 3.
To summarize, when you use synthetic division on the polynomial x4 + x + 1 with r = 1, you find that:
- The quotient is x3 + x2 + 2x + 2
- The remainder is 3
If you’re looking for exact roots or would like to explore other potential values for r, you can use numerical methods or other techniques such as the Rational Root Theorem to test more roots systematically. Happy solving!