MathematicsChapter 5: Polynomials
Exam Response (polynomials)
Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k.
Learning objectives
- State the main ideas and notation used in exam response (polynomials).
- Apply the rules of exam response (polynomials) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k. Write p(c) as its own line. If you claim (x-c) is a factor, show remainder 0.
Formula
c=-3 p(x)
Where / symbols
- EXAM RESPONSE (POLYNOMIALS)
- Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k. Write p(c) as its own line. If you claim (x-c) is a factor, show remainder 0.
- 1. REMAINDER / FACTOR
- Identify c from x-c. Compute p(c). Zero means factor. Non-zero is the remainder.
- 2. FACTORISE
- Find one root. Synthetic or long divide. Quadratic formula. Multiplicity if a repeated factor. Vieta to check.
Relationship
- remainder = p(c)
- factor iff p(c)=0
Overview
Substitute, then divide:
Remainder / Factor
Identify c from x-c. Compute p(c). Zero means factor. Non-zero is the remainder.
Examples x+2 -> p(-2) · Show the arithmetic · State remainder
Factorise
Find one root. Synthetic or long divide. Quadratic formula. Multiplicity if a repeated factor. Vieta to check.
Examples Guess 1,-1,2 · Quotient quadratic · Expand check
WHAT — EXAM RESPONSE (POLYNOMIALS)
Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k. Write p(c) as its own line. If you claim (x-c) is a factor, show remainder 0.
RULE — 1. REMAINDER / FACTOR
Identify c from x-c. Compute p(c). Zero means factor. Non-zero is the remainder.
METHOD — 2. FACTORISE
Find one root. Synthetic or long divide. Quadratic formula. Multiplicity if a repeated factor. Vieta to check.
WORKED EXAMPLE
Worked 4-mark remainder. divisor x+3 means c=-3 p(x)=2x^3 - x + 4 p(-3)=2(-27) -(-3) +4 = -54+3+4=-47 remainder -47
How to answer a exam response (polynomials) 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
- Worked 4-mark remainder.
- divisor x+3 means c=-3
- p(x)=2x3 - x + 4
- p(-3)=2(-27) -(-3) +4 = -54+3+4=-47
- remainder -47
In simple words
x+3 is x-(-3). Plug -3, not 3.
Key points
- remainder = p(c)
- factor iff p(c)=0
- c from the divisor
- one root then divide
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Exam Response (polynomials)
Sticky Notes · Quick Revision
Core Idea
- Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k.
- Write p(c) as its own line.
- If you claim (x-c) is a factor, show remainder 0.
1. REMAINDER / FACTOR
- Identify c from x-c.
- Compute p(c).
- Zero means factor.
- Non-zero is the remainder.
2. FACTORISE
- Find one root.
- Synthetic or long divide.
- Quadratic formula.
- Multiplicity if a repeated factor.
Worked Example
- Worked 4-mark remainder.
- divisor x+3 means c=-3
- p(x)=2x3 - x + 4
- p(-3)=2(-27) -(-3) +4 = -54+3+4=-47
- remainder -47
Common Misconception
- x+3 is x-(-3). Plug -3, not 3.
FBISE Exam Check
- State the rule for exam response (polynomials) before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k.
- Write p(c) as its own line.
- If you claim (x-c) is a factor, show remainder 0.
- x+3 is x-(-3). Plug -3, not 3.
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