MathematicsChapter 5: Polynomials
Roots and Coefficients (school Cubic)
For a cubic x^3 + b x^2 + c x + d = 0 with roots p,q,r (monic): p+q+r = -b, pq+qr+rp = c, pqr = -d.
Learning objectives
- State the main ideas and notation used in roots and coefficients (school cubic).
- Apply the rules of roots and coefficients (school cubic) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
For a cubic x^3 + b x^2 + c x + d = 0 with roots p,q,r (monic): p+q+r = -b, pq+qr+rp = c, pqr = -d. For ax^3+... divide through by a first, or use the general Vieta signs. Meaning: you can find sums of roots without finding each root. Useful when the paper gives some roots and asks the others.
Mathematical form
d = 0 with roots p
r = -b
rp = c
pqr = -d. For ax3+... divide through by a fir
Where / symbols
- ROOTS AND COEFFICIENTS (SCHOOL CUBIC)
- For a cubic x^3 + b x^2 + c x + d = 0 with roots p,q,r (monic): p+q+r = -b, pq+qr+rp = c, pqr = -d. For ax^3+... divide through by a first, or use the general Vieta signs. Meaning: you can find sums of roots without finding each root. Useful when the paper gives some roots and asks the others.
- 1. MONIC CUBIC
- x^3 + b x^2 + c x + d = (x-p)(x-q)(x-r). Expand and match. Signs: (x-p) so sum is -b. Write the three identities.
- 2. USE
- If two roots are 1 and 2, and sum is 6, third is 3. Then build the cubic (x-1)(x-2)(x-3). For quadratics you already know sum=-B/A, product=C/A.
Relationship
- sum of roots = -coeff of x^2 (monic)
- product = -constant (monic cubic)
Overview
Vieta, then a missing root:
Monic Cubic
x^3 + b x^2 + c x + d = (x-p)(x-q)(x-r). Expand and match. Signs: (x-p) so sum is -b. Write the three identities.
Examples sum -b · pairs c · product -d
Use
If two roots are 1 and 2, and sum is 6, third is 3. Then build the cubic (x-1)(x-2)(x-3). For quadratics you already know sum=-BA, product=C/A.
Examples sum - known roots · Build factors · Leading a if not monic
WHAT — ROOTS AND COEFFICIENTS (SCHOOL CUBIC)
For a cubic x^3 + b x^2 + c x + d = 0 with roots p,q,r (monic): p+q+r = -b, pq+qr+rp = c, pqr = -d. For ax^3+... divide through by a first, or use the general Vieta signs. Meaning: you can find sums of roots without finding each root. Useful when the paper gives some roots and asks the others.
RULE — 1. MONIC CUBIC
x^3 + b x^2 + c x + d = (x-p)(x-q)(x-r). Expand and match. Signs: (x-p) so sum is -b. Write the three identities.
METHOD — 2. USE
If two roots are 1 and 2, and sum is 6, third is 3. Then build the cubic (x-1)(x-2)(x-3). For quadratics you already know sum=-BA, product=C/A.
WORKED EXAMPLE
Worked: x^3 - 6x^2 + 11x - 6, guess roots 1,2,3. sum 1+2+3=6 = -(-6) product 6 = -(-6) pairs 2+3+6=11, matches
How to answer a roots and coefficients (school cubic) 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
- x^3 - 6x^2 + 11x - 6, guess roots 1,2,3.
- sum 1+2+3=6 = -(-6)
- product 6 = -(-6)
- pairs 2+3+6=11, matches
In simple words
These identities need the polynomial written with leading 1, or you divide by a first.
Key points
- sum of roots = -coeff of x^2 (monic)
- product = -constant (monic cubic)
- match expansion
- one unknown root
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Roots and Coefficients (school Cubic)
Sticky Notes · Quick Revision
Core Idea
- For a cubic x^3 + b x^2 + c x + d = 0 with roots p,q,r (monic): p+q+r = -b, pq+qr+rp = c, pqr = -d.
- For ax^3+...
- divide through by a first, or use the general Vieta signs.
1. MONIC CUBIC
- x3 + b x2 + c x + d = (x-p)(x-q)(x-r).
- Expand and match.
- Signs: (x-p) so sum is -b.
- Write the three identities.
2. USE
- If two roots are 1 and 2, and sum is 6, third is 3.
- Then build the cubic (x-1)(x-2)(x-3).
- For quadratics you already know sum=-BA, product=C/A.
Worked Example
- Worked: x^3 - 6x^2 + 11x - 6, guess roots 1,2,3.
- sum 1+2+3=6 = -(-6)
- product 6 = -(-6)
- pairs 2+3+6=11, matches
Common Misconception
- These identities need the polynomial written with leading 1, or you divide by a first.
FBISE Exam Check
- State the rule for roots and coefficients (school cubic) before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- For a cubic x^3 + b x^2 + c x + d = 0 with roots p,q,r (monic): p+q+r = -b, pq+qr+rp = c, pqr = -d.
- For ax^3+...
- divide through by a first, or use the general Vieta signs.
- These identities need the polynomial written with leading 1, or you divide by a first.
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