Synthetic division is a simplified method of polynomial long division, allowing us to divide a polynomial by a linear factor in a more straightforward way. Let’s break down the process step by step for the polynomial 2x³ + 0x² + 3x + 22, assuming we want to divide it by x – r, where r is the root we are dividing by.
1. **Identify the coefficients**: In our polynomial 2x³ + 0x² + 3x + 22, the coefficients are 2, 0, 3, and 22.
2. **Select the root (r)**: For example, let’s divide by x – 2, so r = 2.
3. **Set up synthetic division**: Write the coefficients in a row and the value of r to the left:
2 0 3 22 | 2
4. **Bring down the leading coefficient**: Bring down the first coefficient (2):
2 0 3 22 | 2 --------- 2
5. **Multiply and add**: Multiply r (2) by the number directly below it (2), giving us 4. Write this under the next coefficient (0) and add:
2 0 3 22 | 2 --------- 2 4
2 0 3 22 | 2 --------- 2 4 0 + 4 = 4
Continue this process:
2 0 3 22 | 2 --------- 2 4 8 0 + 4 = 4 3 + 8 = 11
2 0 3 22 | 2 --------- 2 4 11 3 + 8 = 11 22 + 22 = 44
Now we have:
2 0 3 22 | 2 --------- 2 4 11 44
6. **Write the result**: The last number (44) is the remainder, while the numbers above represent the quotient. Therefore, the quotient of 2x³ + 3x + 22 divided by x – 2 is 2x² + 4x + 11 with a remainder of 44.
In concluding, using synthetic division makes it easier to find the quotient and remainder of polynomial division efficiently. Understanding this method aids in various algebraic applications, indeed simplifying the process for upper-level math problems.