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After a Root
Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear factors if all roots real. Irreducible quadratic may stay.
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After a Root
What is it?
Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear factors if all roots real. Irreducible quadratic may stay.
In simple words
- Polynomial divide (or synthetic) by x-c.
- Cubic -> linear * quadratic
- Then quadratic formula
- State the rule, show the working, and check the answer.
Important terms
AFTER A ROOT: Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear factors if all roots real. Irreducible quadratic may stay.
AFTER A ROOT: Polynomial divide (or synthetic) by x-c.
What you should learn
- State the rule or meaning of after a root in easy words.
- Apply after a root in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Examples
- Cubic -> linear * quadratic
- Then quadratic formula
- Write p(x)=leading (x-r)(x-s)(x-t)
MAIN IDEA
Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear factors if all roots real. Irreducible quadratic may stay.
EXAM USE
Write the rule first: Polynomial divide (or synthetic) by x-c. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Recommended for Class 11
Full school-level explanation
Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear factors if all roots real. Irreducible quadratic may stay.
Understand the idea, then apply it:
WHAT — AFTER A ROOT
Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear factors if all roots real. Irreducible quadratic may stay.
WHY — Why it matters
This rule is a separate learning unit inside factor theorem. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: Polynomial divide (or synthetic) by x-c. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
MAIN IDEA
Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear factors if all roots real. Irreducible quadratic may stay.
Remember
- divide out x-c
Examples
- Cubic -> linear * quadratic
- Then quadratic formula
- Write p(x)=leading (x-r)(x-s)(x-t)
EXAM USE
Write the rule first: Polynomial divide (or synthetic) by x-c. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Remember
- focused answer
- accurate support
Examples
- State the central idea
- Use one accurate detail
- Check the final response
How to answer a after a root question
How it works
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
Remember
- divide out x-c
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
- p(c)=0
- divide out x-c
- focused answer
- accurate support
- divide out x-c
Deeper understanding
Go beyond memorising After a Root. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
Core Idea
What to Know
Example or Evidence
How to Answer
Exam focus
Quick revision
- divide out x-c
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
- p(c)=0
- AFTER A ROOT: Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear fa
- AFTER A ROOT: Polynomial divide (or synthetic) by x-c.
A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
For long answers on After a Root: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- divide out x-c
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
- p(c)=0
Important formulas
- divide out x-c
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
Common exam traps
- A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
- Do not choose a formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
- Mistake
Do not choose a formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes, restrictions, or skipped algebra steps.
- Mistake
A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
divide out x-c
p(c)=0 <-> (x-c) is a factor
guess small integer roots
p(c)=0
Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear factors if all roots real. Irreducible quadratic may stay.
p(c)=0 <-> (x-c) is a factor
- Definition
- Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear factors if all roots real. Irreducible quadratic may stay.
- Formula
- p(c)=0 <-> (x-c) is a factor
- Example
- Cubic -> linear * quadratic
- Critical warning
- A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
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Exam Practice
Define
Q1. Define After a Root.
Explain
Q2. Explain: MAIN IDEA.
Example
Q3. Give one clear example of After a Root and say why it fits.
Mistake
Q4. State one common mistake students make about After a Root.
Exam note
Q5. Write a short exam note on After a Root: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.