Polynomial division appears in the Advanced Math section of the SAT. You'll either divide polynomials directly or use the Remainder Theorem as a shortcut. Here's how to handle both.
Polynomial Long Division
Long division with polynomials works just like long division with numbers.
Example 1: Divide by
Step 1: Divide the leading terms:
Step 2: Multiply:
Step 3: Subtract:
Step 4: Bring down:
Step 5: Repeat: , then
Step 6: Subtract:
Step 7: Bring down:
Step 8: Repeat: , then
Step 9: Subtract:
Synthetic Division
Synthetic division is a shortcut that works when dividing by .
Example 2: Divide by
Use . Write the coefficients:
| 1 | -4 | 6 | -4 | |
|---|---|---|---|---|
| Bring down | 1 | |||
| Multiply by 2 | 2 | |||
| Add | 1 | -2 | ||
| Multiply by 2 | -4 | |||
| Add | 1 | -2 | 2 | |
| Multiply by 2 | 4 | |||
| Add | 1 | -2 | 2 | 0 |
Result: with remainder .
Since the remainder is 0, is a factor.
The Remainder Theorem
This is the SAT's favorite shortcut:
When you divide by , the remainder equals .
Example 3: What is the remainder when is divided by ?
Just plug in :
The remainder is . No division needed.
The Factor Theorem
If , then is a factor of .
Example 4: Is a factor of ?
✓
Yes, is a factor.
Practice Problems
Problem 1: What is the remainder when is divided by ?
Solution
. Remainder is 3.
Problem 2: Is a factor of ?
Solution
. Yes, it's a factor.
Problem 3: Divide by using synthetic division.
Solution
Use , coefficients:
Result: with remainder 0. So .
Key Takeaways
- Polynomial long division follows the same steps as numerical long division
- Synthetic division is faster but only works for divisors of the form
- Remainder Theorem: the remainder of is just
- Factor Theorem: if , then is a factor
- On the SAT, try the Remainder Theorem first — it's often all you need
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