MathematicsChapter 5: Polynomials
After A Root
Polynomial divide (or synthetic) by x-c.
Learning objectives
- 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.
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.
Where / symbols
- 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.
Relationship
- p(c)=0 <-> (x-c) is a factor
Overview
Understand the idea, then apply it:
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.
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.
Examples State the central idea · Use one accurate detail · Check the final response
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.
How to answer a after a root question
- 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.
In simple words
A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
Key points
- divide out x-c
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
- focused answer
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.After A Root
Sticky Notes · Quick Revision
Core Idea
- Polynomial divide (or synthetic) by x-c.
What to Know
- Polynomial divide (or synthetic) by x-c. Quadratic formula or inspect the quadratic. Three linear factors if all roots real. Irreducible quadratic may stay.
Example or Evidence
- Cubic -> linear * quadratic
- Then quadratic formula
- Write p(x)=leading (x-r)(x-s)(x-t)
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.
Common Error
- A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
FBISE Exam Check
- State the rule, show the working, and check the answer.
60-second revision
- Polynomial divide (or synthetic) by x-c.
- Cubic -> linear * quadratic
- Then quadratic formula
- State the rule, show the working, and check the answer.
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