MathematicsChapter 5: Polynomials
Factor Theorem
x-c is a factor of p(x) iff p(c)=0.
Learning objectives
- 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.
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), then divide to get a quadratic, then factor that. Meaning: a root gives a linear factor, and the other way around.
Where / symbols
- 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.
Relationship
- p(c)=0 <-> (x-c) is a factor
Overview
Test a value, then split:
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).
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.
Examples Cubic -> linear * quadratic · Then quadratic formula · Write p(x)=leading (x-r)(x-s)(x-t)
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
How to answer a factor theorem question
- 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.
Worked example
- factor x3 - 3x - 2 if p(2)=0.
- p(2)=8-6-2=0 so (x-2) is a factor
- divide: x3-3x-2 = (x-2)(x2+2x+1)
- x2+2x+1=(x+1)2
- = (x-2)(x+1)^2
In simple words
A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
Key points
- p(c)=0 <-> (x-c) is a factor
- guess small integer roots
- plug in
- divide out x-c
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Factor Theorem
Sticky Notes · Quick Revision
Core Idea
- 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.
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.
Worked Example
- Worked: factor x3 - 3x - 2 if p(2)=0.
- p(2)=8-6-2=0 so (x-2) is a factor
- divide: x3-3x-2 = (x-2)(x2+2x+1)
- x2+2x+1=(x+1)2
- = (x-2)(x+1)^2
Common Misconception
- A factor x-c needs p(c)=0, not p(-c)=0 unless c is negative.
FBISE Exam Check
- State the rule for factor theorem before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.