MathematicsChapter 5: Polynomials
Zeros and A Cubic Worked Pattern
A zero of p is a root: p(c)=0.
Learning objectives
- State the main ideas and notation used in zeros and a cubic worked pattern.
- Apply the rules of zeros and a cubic worked pattern in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
A zero of p is a root: p(c)=0. Counting multiplicity: (x-1)^2 has a double zero at 1. A cubic has three zeros in C, counting multiplicity (school: maybe one real and two conjugate, or three real). Graph: it crosses or touches the x-axis at real zeros. Meaning: factorisation is a list of zeros with repeats.
Mathematical form
x=1 with multiplicity 2
x=-4 with multiplicity 1
x=1 with multiplicity 2 x
Where / symbols
- ZEROS AND A CUBIC WORKED PATTERN
- A zero of p is a root: p(c)=0. Counting multiplicity: (x-1)^2 has a double zero at 1. A cubic has three zeros in C, counting multiplicity (school: maybe one real and two conjugate, or three real). Graph: it crosses or touches the x-axis at real zeros. Meaning: factorisation is a list of zeros with repeats.
- 1. MULTIPLICITY
- Touch and turn: even multiplicity (school cubic can have a repeated factor). Cross: odd multiplicity. Sum of multiplicities = degree.
- 2. PLAN FOR CUBIC
- 1) try p(1), p(-1), ... 2) factor theorem 3) divide 4) quadratic. Show p(c)=0 as a line of its own. Remainder 0 is the check.
Relationship
- zero = root = x-intercept if real
- multiplicity = power of the factor
Overview
List zeros, then a cubic plan:
Multiplicity
Touch and turn: even multiplicity (school cubic can have a repeated factor). Cross: odd multiplicity. Sum of multiplicities = degree.
Examples Double root · Triple root cubic (x-c)^3 · Complex zeros pair if real coeffs
Plan For Cubic
1) try p(1), p(-1), ... 2) factor theorem 3) divide 4) quadratic. Show p(c)=0 as a line of its own. Remainder 0 is the check.
Examples Guess · Divide · Quadratic · Write factors
WHAT — ZEROS AND A CUBIC WORKED PATTERN
A zero of p is a root: p(c)=0. Counting multiplicity: (x-1)^2 has a double zero at 1. A cubic has three zeros in C, counting multiplicity (school: maybe one real and two conjugate, or three real). Graph: it crosses or touches the x-axis at real zeros. Meaning: factorisation is a list of zeros with repeats.
RULE — 1. MULTIPLICITY
Touch and turn: even multiplicity (school cubic can have a repeated factor). Cross: odd multiplicity. Sum of multiplicities = degree.
METHOD — 2. PLAN FOR CUBIC
1) try p(1), p(-1), ... 2) factor theorem 3) divide 4) quadratic. Show p(c)=0 as a line of its own. Remainder 0 is the check.
WORKED EXAMPLE
Worked: zeros of (x-1)^2 (x+4). x=1 with multiplicity 2 x=-4 with multiplicity 1 degree 3, three zeros counted with repeats
How to answer a zeros and a cubic worked pattern 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
- zeros of (x-1)^2 (x+4).
- x=1 with multiplicity 2
- x=-4 with multiplicity 1
- degree 3, three zeros counted with repeats
In simple words
Three zeros counted with multiplicity, not always three different numbers.
Key points
- zero = root = x-intercept if real
- multiplicity = power of the factor
- power on (x-c)
- four steps
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Zeros and A Cubic Worked Pattern
Sticky Notes · Quick Revision
Core Idea
- A zero of p is a root: p(c)=0.
- Counting multiplicity: (x-1)^2 has a double zero at 1.
- A cubic has three zeros in C, counting multiplicity (school: maybe one real and two conjugate, or three real).
1. MULTIPLICITY
- Touch and turn: even multiplicity (school cubic can have a repeated factor).
- Cross: odd multiplicity.
- Sum of multiplicities = degree.
2. PLAN FOR CUBIC
- 1) try p(1), p(-1), ...
- 2) factor theorem 3) divide 4) quadratic.
- Show p(c)=0 as a line of its own.
- Remainder 0 is the check.
Worked Example
- Worked: zeros of (x-1)^2 (x+4).
- x=1 with multiplicity 2
- x=-4 with multiplicity 1
- degree 3, three zeros counted with repeats
Common Misconception
- Three zeros counted with multiplicity, not always three different numbers.
FBISE Exam Check
- State the rule for zeros and a cubic worked pattern before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- A zero of p is a root: p(c)=0.
- Counting multiplicity: (x-1)^2 has a double zero at 1.
- A cubic has three zeros in C, counting multiplicity (school: maybe one real and two conjugate, or three real).
- Three zeros counted with multiplicity, not always three different numbers.
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