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Synthetic Division (school)
Synthetic division is a short table for dividing by x-c. Write c on the left, coefficients in a row (include 0 for missing powers). Bring down, multiply by c, add, repeat. Last number is the remainder (and p(c)). The other numbers are coefficients of the quotient, degree one less. Meaning: same arithmetic as long division, less writing.
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Synthetic Division (school)
What is it?
Synthetic division is a short table for dividing by x-c. Write c on the left, coefficients in a row (include 0 for missing powers). Bring down, multiply by c, add, repeat. Last number is the remainder (and p(c)). The other numbers are coefficients of the quotient, degree one less. Meaning: same arithmetic as long division, less writing.
In simple words
- Synthetic division is a short table for dividing by x-c.
- Write c on the left, coefficients in a row (include 0 for missing powers).
- Bring down, multiply by c, add, repeat.
- For x+3, the left number is -3, not +3.
Important terms
SYNTHETIC DIVISION (SCHOOL): Synthetic division is a short table for dividing by x-c. Write c on the left, coefficients in a row (include 0 for missing powers). Bring down, multiply by c, add, repeat. Last number is the remainder (and p(c)). The other numbers are coefficients of the quotient, degree one less. Meaning: same arithmetic as long division, less writing.
1. TABLE: Missing x^2 in a cubic: write 0. Signs matter. If dividing by x+2, c=-2. Bring down the first coefficient unchanged.
2. QUOTIENT: If p was degree 3, quotient is degree 2 with those first leftover numbers. Then continue factoring the quadratic. Check by expanding back if marks allow.
What you should learn
- State the main ideas and notation used in synthetic division (school).
- Apply the rules of synthetic division (school) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- x-2 -> c=2
- x+1 -> c=-1
- Last = remainder
TABLE
Missing x^2 in a cubic: write 0. Signs matter. If dividing by x+2, c=-2. Bring down the first coefficient unchanged.
QUOTIENT
If p was degree 3, quotient is degree 2 with those first leftover numbers. Then continue factoring the quadratic. Check by expanding back if marks allow.
Recommended for Class 11
Full school-level explanation
Synthetic division is a short table for dividing by x-c. Write c on the left, coefficients in a row (include 0 for missing powers). Bring down, multiply by c, add, repeat. Last number is the remainder (and p(c)). The other numbers are coefficients of the quotient, degree one less. Meaning: same arithmetic as long division, less writing.
Set the row, then read the answer:
WHAT — SYNTHETIC DIVISION (SCHOOL)
Synthetic division is a short table for dividing by x-c. Write c on the left, coefficients in a row (include 0 for missing powers). Bring down, multiply by c, add, repeat. Last number is the remainder (and p(c)). The other numbers are coefficients of the quotient, degree one less. Meaning: same arithmetic as long division, less writing.
RULE — 1. TABLE
Missing x^2 in a cubic: write 0. Signs matter. If dividing by x+2, c=-2. Bring down the first coefficient unchanged.
METHOD — 2. QUOTIENT
If p was degree 3, quotient is degree 2 with those first leftover numbers. Then continue factoring the quadratic. Check by expanding back if marks allow.
WORKED EXAMPLE
Worked: divide x^3 + 0 x^2 - 3x - 2 by x-2 (from earlier). c=2, coeffs 1 0 -3 -2 bring 1; 2 add -> 2; 2 add -> 1; *2 add -> 0 quotient x^2 + 2x + 1, remainder 0
TABLE
Missing x^2 in a cubic: write 0. Signs matter. If dividing by x+2, c=-2. Bring down the first coefficient unchanged.
Remember
- include zeros
Examples
- x-2 -> c=2
- x+1 -> c=-1
- Last = remainder
QUOTIENT
If p was degree 3, quotient is degree 2 with those first leftover numbers. Then continue factoring the quadratic. Check by expanding back if marks allow.
Remember
- one degree less
Examples
- Numbers are coeffs
- Remainder right end
- p(c) check
How to answer a synthetic division (school) question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- c on the left, coeffs in a row
- last output = remainder = p(c)
- remainder 0
- c is from x-c
- include zeros
- one degree less
- c on the left, coeffs in a row
- last output = remainder = p(c)
How the ideas connect
Worked: divide x^3 + 0 x^2 - 3x - 2 by x-2 (from earlier). c=2, coeffs 1 0 -3 -2 bring 1; 2 add -> 2; 2 add -> 1; *2 add -> 0 quotient x^2 + 2x + 1, remainder 0
Deeper understanding
Go beyond memorising Synthetic Division (school). Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: divide x^3 + 0 x^2 - 3x - 2 by x-2 (from earlier). c=2, coeffs 1 0 -3 -2 bring 1; 2 add -> 2; 2 add -> 1; *2 add -> 0 quotient x^2 + 2x + 1, remainder 0
Common mistakes
Exam focus
Core Idea
1. TABLE
2. QUOTIENT
Worked Example
Exam focus
Quick revision
- c on the left, coeffs in a row
- last output = remainder = p(c)
- remainder 0
- c is from x-c
- SYNTHETIC DIVISION (SCHOOL): Synthetic division is a short table for dividing by x-c. Write c on the left, coefficients in a row
- 1. TABLE: Missing x^2 in a cubic: write 0. Signs matter. If dividing by x+2, c=-2. Bring down the first coeffi
For x+3, the left number is -3, not +3.
For long answers on Synthetic Division (school): define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- c on the left, coeffs in a row
- last output = remainder = p(c)
- remainder 0
- c is from x-c
Important formulas
- c on the left, coeffs in a row
- last output = remainder = p(c)
- remainder 0
Common exam traps
- For x+3, the left number is -3, not +3.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
For x+3, the left number is -3, not +3.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
For x+3, the left number is -3, not +3.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
c on the left, coeffs in a row
last output = remainder = p(c)
remainder 0
c is from x-c
Synthetic division is a short table for dividing by x-c. Write c on the left, coefficients in a row (include 0 for missing powers). Bring down, multiply by c, a
last output = remainder = p(c)
- Definition
- Synthetic division is a short table for dividing by x-c. Write c on the left, coefficients in a row (include 0 for missing powers). Bring down, multiply by c, a
- Formula
- last output = remainder = p(c)
- Example
- x-2 -> c=2
- Critical warning
- For x+3, the left number is -3, not +3.
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Exam Practice
Define
Q1. Define Synthetic Division (school).
Explain
Q2. Explain: TABLE.
Example
Q3. Give one clear example of Synthetic Division (school) and say why it fits.
Mistake
Q4. State one common mistake students make about Synthetic Division (school).
Exam note
Q5. Write a short exam note on Synthetic Division (school): definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.