Synthetic division - by a linear divisor
Divide a polynomial by
Optional method: synthetic division is not required by the AP Precalculus exam. Polynomial long division covers the required division skill.
- Write the coefficients in descending powers, including a
for every missing power. - Use the zero of the divisor:
for , or for . - Bring down the first coefficient. Multiply it by
, write the product under the next coefficient, and add. Repeat across the row. - The last number is the remainder
. The preceding numbers are the quotient coefficients, starting one degree lower.
Example 1
An exact division
Divide
| Use root | ||||
|---|---|---|---|---|
| Multiply by the root | Bring down | |||
| Add down each column | ||||
| Quotient coefficients | Remainder | |||
Read the quotient one degree lower. The last number is the remainder.
The quotient has degree
Example 2
Reorder first and include a missing power
Divide
Write descending powers and insert
| Use root | |||||
|---|---|---|---|---|---|
| Multiply by the root | Bring down | ||||
| Add down each column | |||||
| Quotient coefficients | Remainder | ||||
Read the quotient one degree lower. The last number is the remainder.
The quotient has degree
Remember
Check with
Your turn
Use synthetic division. State the quotient and remainder.
Check answer
| Use root | ||||
|---|---|---|---|---|
| Multiply by the root | Bring down | |||
| Add down each column | ||||
| Quotient coefficients | Remainder | |||
Read the quotient one degree lower. The last number is the remainder.
Use synthetic division. State the quotient and remainder.
Check answer
| Use root | ||||
|---|---|---|---|---|
| Multiply by the root | Bring down | |||
| Add down each column | ||||
| Quotient coefficients | Remainder | |||
Read the quotient one degree lower. The last number is the remainder.
Use synthetic division. State the quotient and remainder.
Check answer
| Use root | ||||
|---|---|---|---|---|
| Multiply by the root | Bring down | |||
| Add down each column | ||||
| Quotient coefficients | Remainder | |||
Read the quotient one degree lower. The last number is the remainder.
Use synthetic division. State the quotient and remainder.
Check answer
| Use root | ||||
|---|---|---|---|---|
| Multiply by the root | Bring down | |||
| Add down each column | ||||
| Quotient coefficients | Remainder | |||
Read the quotient one degree lower. The last number is the remainder.
Use synthetic division. State the quotient and remainder.
Check answer
| Use root | |||||
|---|---|---|---|---|---|
| Multiply by the root | Bring down | ||||
| Add down each column | |||||
| Quotient coefficients | Remainder | ||||
Read the quotient one degree lower. The last number is the remainder.
Use synthetic division. State the quotient and remainder.
| Use root | ||||
|---|---|---|---|---|
| Multiply by the root | Bring down | |||
| Add down each column | ||||
| Quotient coefficients | Remainder | |||
Read the quotient one degree lower. The last number is the remainder.
Use synthetic division. State the quotient and remainder.
| Use root | ||||
|---|---|---|---|---|
| Multiply by the root | Bring down | |||
| Add down each column | ||||
| Quotient coefficients | Remainder | |||
Read the quotient one degree lower. The last number is the remainder.
Use synthetic division. State the quotient and remainder.
| Use root | ||||
|---|---|---|---|---|
| Multiply by the root | Bring down | |||
| Add down each column | ||||
| Quotient coefficients | Remainder | |||
Read the quotient one degree lower. The last number is the remainder.
Use synthetic division. State the quotient and remainder.
| Use root | ||||
|---|---|---|---|---|
| Multiply by the root | Bring down | |||
| Add down each column | ||||
| Quotient coefficients | Remainder | |||
Read the quotient one degree lower. The last number is the remainder.
Use synthetic division. State the quotient and remainder.
| Use root | |||||
|---|---|---|---|---|---|
| Multiply by the root | Bring down | ||||
| Add down each column | |||||
| Quotient coefficients | Remainder | ||||
Read the quotient one degree lower. The last number is the remainder.