MathematicsChapter 5: Polynomials
Polynomials and Division
A polynomial in x is a_n x^n + ...
Learning objectives
- State the main ideas and notation used in polynomials and division.
- Apply the rules of polynomials and division in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
A polynomial in x is a_n x^n + ... + a1 x + a0 with real (or complex) coefficients, n a whole number. Degree is the highest power with a non-zero coefficient. Constant non-zero: degree 0. Zero polynomial: degree left undefined or -infinity in some books. Division: p(x) = q(x) d(x) + r(x) with deg r < deg d (or r=0). Meaning: like 17 = 5*3 + 2, but with expressions.
Formula
r=0). Meaning
Where / symbols
- POLYNOMIALS AND DIVISION
- A polynomial in x is a_n x^n + ... + a1 x + a0 with real (or complex) coefficients, n a whole number. Degree is the highest power with a non-zero coefficient. Constant non-zero: degree 0. Zero polynomial: degree left undefined or -infinity in some books. Division: p(x) = q(x) d(x) + r(x) with deg r < deg d (or r=0). Meaning: like 17 = 5*3 + 2, but with expressions.
- 1. DEGREE AND ZEROS
- Cubic degree 3. Leading coefficient a_n. A zero (root) is a number c with p(c)=0. Factor theorem next. Do not call 1/x a polynomial.
- 2. DIVISION ALGORITHM
- Divide by a linear x-c: remainder is a constant. Long division or synthetic (later). Stop when the leftover degree is less than the divisor's degree.
Relationship
- degree = highest power
Overview
What it is, then divide:
Degree and Zeros
Cubic degree 3. Leading coefficient a_n. A zero (root) is a number c with p(c)=0. Factor theorem next. Do not call 1x a polynomial.
Examples x^3-1 degree 3 · 7 degree 0 · p(c)=0 is a root
Division Algorithm
Divide by a linear x-c: remainder is a constant. Long division or synthetic (later). Stop when the leftover degree is less than the divisor's degree.
Examples Like integers · Linear divisor: r = number · p = q d + r
WHAT — POLYNOMIALS AND DIVISION
A polynomial in x is a_n x^n + ... + a1 x + a0 with real (or complex) coefficients, n a whole number. Degree is the highest power with a non-zero coefficient. Constant non-zero: degree 0. Zero polynomial: degree left undefined or -infinity in some books. Division: p(x) = q(x) d(x) + r(x) with deg r < deg d (or r=0). Meaning: like 17 = 5*3 + 2, but with expressions.
RULE — 1. DEGREE AND ZEROS
Cubic degree 3. Leading coefficient a_n. A zero (root) is a number c with p(c)=0. Factor theorem next. Do not call 1x a polynomial.
METHOD — 2. DIVISION ALGORITHM
Divide by a linear x-c: remainder is a constant. Long division or synthetic (later). Stop when the leftover degree is less than the divisor's degree.
WORKED EXAMPLE
Worked: degree of 4x^2 - x^5 + 2. highest power is 5 degree 5 leading coefficient -1
How to answer a polynomials and division 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
- degree of 4x^2 - x^5 + 2.
- highest power is 5
- degree 5
- leading coefficient -1
In simple words
1x and sqrt(x) are not polynomials. Powers must be 0,1,2,...
Key points
- degree = highest power
- divide: quotient + remainder, deg r smaller
- whole powers only
- r smaller degree
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Polynomials and Division
Sticky Notes · Quick Revision
Core Idea
- A polynomial in x is a_n x^n + ...
- + a1 x + a0 with real (or complex) coefficients, n a whole number.
- Degree is the highest power with a non-zero coefficient.
1. DEGREE AND ZEROS
- Cubic degree 3.
- Leading coefficient a_n.
- A zero (root) is a number c with p(c)=0.
- Factor theorem next.
2. DIVISION ALGORITHM
- Divide by a linear x-c: remainder is a constant.
- Long division or synthetic (later).
- Stop when the leftover degree is less than the divisor's degree.
Worked Example
- Worked: degree of 4x^2 - x^5 + 2.
- highest power is 5
- degree 5
- leading coefficient -1
Common Misconception
- 1x and sqrt(x) are not polynomials. Powers must be 0,1,2,...
FBISE Exam Check
- State the rule for polynomials and division before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- A polynomial in x is a_n x^n + ...
- a1 x + a0 with real (or complex) coefficients, n a whole number.
- Degree is the highest power with a non-zero coefficient.
- 1x and sqrt(x) are not polynomials. Powers must be 0,1,2,...
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