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Factor Theorem
x-c is a factor of p(x) iff p(c)=0. That is the remainder theorem with remainder 0. To factor a cubic, guess a simple root (factors of constant term, school), then divide to get a quadratic, then factor that. Meaning: a root gives a linear factor, and the other way around.
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Factor Theorem
What is it?
x-c is a factor of p(x) iff p(c)=0. That is the remainder theorem with remainder 0. To factor a cubic, guess a simple root (factors of constant term, school), then divide to get a quadratic, then factor that. Meaning: a root gives a linear factor, and the other way around.
In simple words
- x-c is a factor of p(x) iff p(c)=0.
- That is the remainder theorem with remainder 0.
- To factor a cubic, guess a simple root (factors of constant term, school), then divide to get a quadratic, then factor that.
- A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
Important terms
FACTOR THEOREM: x-c is a factor of p(x) iff p(c)=0. That is the remainder theorem with remainder 0. To factor a cubic, guess a simple root (factors of constant term, school), then divide to get a quadratic, then factor that. Meaning: a root gives a linear factor, and the other way around.
1. TEST: Try x=1, -1, 2, -2 on a cubic with integer coeffs. If p(1)=0, (x-1) divides. Integer-root guess: factors of a0 over factors of a_n (rational root idea named).
2. 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.
What you should learn
- State the main ideas and notation used in factor theorem.
- Apply the rules of factor theorem in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- p(1)=0 -> factor x-1
- p(-2)=0 -> x+2
- No luck: maybe no integer root
TEST
Try x=1, -1, 2, -2 on a cubic with integer coeffs. If p(1)=0, (x-1) divides. Integer-root guess: factors of a0 over factors of a_n (rational root idea named).
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.
Recommended for Class 11
Full school-level explanation
x-c is a factor of p(x) iff p(c)=0. That is the remainder theorem with remainder 0. To factor a cubic, guess a simple root (factors of constant term, school), then divide to get a quadratic, then factor that. Meaning: a root gives a linear factor, and the other way around.
Test a value, then split:
WHAT — FACTOR THEOREM
x-c is a factor of p(x) iff p(c)=0. That is the remainder theorem with remainder 0. To factor a cubic, guess a simple root (factors of constant term, school), then divide to get a quadratic, then factor that. Meaning: a root gives a linear factor, and the other way around.
RULE — 1. TEST
Try x=1, -1, 2, -2 on a cubic with integer coeffs. If p(1)=0, (x-1) divides. Integer-root guess: factors of a0 over factors of a_n (rational root idea named).
METHOD — 2. 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.
WORKED EXAMPLE
Worked: factor x^3 - 3x - 2 if p(2)=0. p(2)=8-6-2=0 so (x-2) is a factor divide: x^3-3x-2 = (x-2)(x^2+2x+1) x^2+2x+1=(x+1)^2 = (x-2)(x+1)^2
TEST
Try x=1, -1, 2, -2 on a cubic with integer coeffs. If p(1)=0, (x-1) divides. Integer-root guess: factors of a0 over factors of a_n (rational root idea named).
Remember
- plug in
Examples
- p(1)=0 -> factor x-1
- p(-2)=0 -> x+2
- No luck: maybe no integer root
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.
Remember
- divide out x-c
Examples
- Cubic -> linear * quadratic
- Then quadratic formula
- Write p(x)=leading (x-r)(x-s)(x-t)
How to answer a factor theorem 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
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
- (x-2)(x+1)^2
- p(c)=0
- plug in
- divide out x-c
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
How the ideas connect
Worked: factor x^3 - 3x - 2 if p(2)=0. p(2)=8-6-2=0 so (x-2) is a factor divide: x^3-3x-2 = (x-2)(x^2+2x+1) x^2+2x+1=(x+1)^2 = (x-2)(x+1)^2
Deeper understanding
Go beyond memorising Factor Theorem. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: factor x^3 - 3x - 2 if p(2)=0. p(2)=8-6-2=0 so (x-2) is a factor divide: x^3-3x-2 = (x-2)(x^2+2x+1) x^2+2x+1=(x+1)^2 = (x-2)(x+1)^2
Common mistakes
Exam focus
Core Idea
1. TEST
2. AFTER A ROOT
Worked Example
Exam focus
Quick revision
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
- (x-2)(x+1)^2
- p(c)=0
- FACTOR THEOREM: x-c is a factor of p(x) iff p(c)=0. That is the remainder theorem with remainder 0. To factor a cubi
- 1. TEST: Try x=1, -1, 2, -2 on a cubic with integer coeffs. If p(1)=0, (x-1) divides. Integer-root guess: fac
A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
For long answers on Factor Theorem: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
- (x-2)(x+1)^2
- p(c)=0
Important formulas
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
- (x-2)(x+1)^2
Common exam traps
- A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
- 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
A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
- 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
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.
p(c)=0 <-> (x-c) is a factor
guess small integer roots
(x-2)(x+1)^2
p(c)=0
x-c is a factor of p(x) iff p(c)=0. That is the remainder theorem with remainder 0. To factor a cubic, guess a simple root (factors of constant term, school), t
p(c)=0 <-> (x-c) is a factor
- Definition
- x-c is a factor of p(x) iff p(c)=0. That is the remainder theorem with remainder 0. To factor a cubic, guess a simple root (factors of constant term, school), t
- Formula
- p(c)=0 <-> (x-c) is a factor
- Example
- p(1)=0 -> factor x-1
- 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 Factor Theorem.
Explain
Q2. Explain: TEST.
Example
Q3. Give one clear example of Factor Theorem and say why it fits.
Mistake
Q4. State one common mistake students make about Factor Theorem.
Exam note
Q5. Write a short exam note on Factor Theorem: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.