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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.
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Zeros and a Cubic Worked Pattern
What is it?
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.
In simple words
- 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.
Important terms
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.
What you should learn
- 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.
Examples
- Double root
- Triple root cubic (x-c)^3
- Complex zeros pair if real coeffs
MULTIPLICITY
Touch and turn: even multiplicity (school cubic can have a repeated factor). Cross: odd multiplicity. Sum of multiplicities = degree.
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.
Recommended for Class 11
Full school-level explanation
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.
List zeros, then a cubic plan:
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
MULTIPLICITY
Touch and turn: even multiplicity (school cubic can have a repeated factor). Cross: odd multiplicity. Sum of multiplicities = degree.
Remember
- power on (x-c)
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.
Remember
- four steps
Examples
- Guess
- Divide
- Quadratic
- Write factors
How to answer a zeros and a cubic worked pattern 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
- zero = root = x-intercept if real
- multiplicity = power of the factor
- 1,1,-4
- count repeats
- power on (x-c)
- four steps
- zero = root = x-intercept if real
- multiplicity = power of the factor
How the ideas connect
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
Deeper understanding
Go beyond memorising Zeros and a Cubic Worked Pattern. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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 mistakes
Exam focus
Core Idea
1. MULTIPLICITY
2. PLAN FOR CUBIC
Worked Example
Exam focus
Quick revision
- zero = root = x-intercept if real
- multiplicity = power of the factor
- 1,1,-4
- count repeats
- 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 th
- 1. MULTIPLICITY: Touch and turn: even multiplicity (school cubic can have a repeated factor). Cross: odd multiplicity
Three zeros counted with multiplicity, not always three different numbers.
For long answers on Zeros and a Cubic Worked Pattern: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- zero = root = x-intercept if real
- multiplicity = power of the factor
- 1,1,-4
- count repeats
Important formulas
- zero = root = x-intercept if real
- multiplicity = power of the factor
- 1,1,-4
Common exam traps
- Three zeros counted with multiplicity, not always three different numbers.
- 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
Three zeros counted with multiplicity, not always three different numbers.
- 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
Three zeros counted with multiplicity, not always three different numbers.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
zero = root = x-intercept if real
multiplicity = power of the factor
1,1,-4
count repeats
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 rea
zero = root = x-intercept if real
- 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 rea
- Formula
- zero = root = x-intercept if real
- Example
- Double root
- Critical warning
- Three zeros counted with multiplicity, not always three different numbers.
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Exam Practice
Define
Q1. Define Zeros and a Cubic Worked Pattern.
Explain
Q2. Explain: MULTIPLICITY.
Example
Q3. Give one clear example of Zeros and a Cubic Worked Pattern and say why it fits.
Mistake
Q4. State one common mistake students make about Zeros and a Cubic Worked Pattern.
Exam note
Q5. Write a short exam note on Zeros and a Cubic Worked Pattern: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.