Cheat Sheet — Polynomials

📐 Types of Polynomials

Type	Degree	General Form	Max Zeroes
Constant	0	k (k ≠ 0)	0
Linear	1	ax + b	1
Quadratic	2	ax² + bx + c	2
Cubic	3	ax³ + bx² + cx + d	3
Biquadratic	4	ax⁴ + bx³ + cx² + dx + e	4

Key Rule A polynomial of degree n has at most n real zeroes. The leading coefficient a ≠ 0.

Zero polynomial: p(x) = 0. Every real number is a zero; its degree is undefined.

Value at x = k: Substitute x = k into p(x). If p(k) = 0, then k is a zero of p(x).

Degree of product: deg[p(x) × q(x)] = deg p(x) + deg q(x).

📈 Zeroes & Graphical Interpretation

Definition k is a zero of p(x) if p(k) = 0. Geometrically, zeroes are the x-coordinates of the points where the graph crosses or touches the x-axis.

Polynomial	Graph Shape	Possible Zeroes
Linear: ax + b	Straight line	Exactly 1
Quadratic: ax² + bx + c	Parabola	0, 1, or 2
Cubic: ax³ + …	S-shaped curve	1, 2, or 3

a > 0 (quadratic): parabola opens upward ∪. a < 0: opens downward ∩.

Graph crosses x-axis → zero with odd multiplicity (simple root).

Graph only touches x-axis (does not cross) → zero with even multiplicity (repeated root).

🔷 Quadratic: Relationship of Zeroes

For p(x) = ax² + bx + c (a ≠ 0) Let α and β be the two zeroes. Then:

SUM OF ZEROES

α + β = −b/a = − (coeff. of x) / (coeff. of x²)

PRODUCT OF ZEROES

αβ = c/a = (constant term) / (coeff. of x²)

FORMING A QUADRATIC FROM ZEROES

p(x) = x² − (α+β)x + αβ

Or: k[x² − (Sum)x + Product] for any non-zero constant k.

Example: p(x) = 2x² − 5x + 3 → α+β = 5/2, αβ = 3/2.

Identity: α² + β² = (α+β)² − 2αβ = (Sum)² − 2(Product)

Identity: (α − β)² = (α+β)² − 4αβ = (Sum)² − 4(Product)

🔮 Cubic: Relationship of Zeroes

For p(x) = ax³ + bx² + cx + d (a ≠ 0) Let α, β, γ be the three zeroes. Then:

SUM OF ZEROES

α + β + γ = −b/a

SUM OF PRODUCTS OF PAIRS

αβ + βγ + γα = c/a

PRODUCT OF ZEROES

αβγ = −d/a

Sign pattern — alternate: −b/a, +c/a, −d/a. Starts with minus.

Example: x³ − 6x² + 11x − 6 → α+β+γ = 6, αβ+βγ+γα = 11, αβγ = 6. (Zeroes: 1, 2, 3.)

➗ Division Algorithm for Polynomials

Theorem (Division Algorithm) If p(x) and g(x) are polynomials with g(x) ≠ 0, then there exist unique polynomials q(x) and r(x) such that:

DIVISION ALGORITHM

p(x) = g(x) × q(x) + r(x)

where deg r(x) < deg g(x), or r(x) = 0.
p = Dividend, g = Divisor, q = Quotient, r = Remainder.

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Verify given zeroes — divide out the known factor and solve the resulting polynomial.

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Find remaining zeroes of a cubic when one zero is known: divide p(x) by (x − α).

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Find quotient and remainder when dividing one polynomial by another.

Example: Divide 3x³+x²+2x+5 by x²+2x+1 → q(x) = 3x−5, r(x) = 9x+10.

If zero α is known → divide by (x−α) → get quadratic → solve for two more zeroes.

🧮 Key Identities & Derived Expressions

Expression	Formula (in terms of S = α+β, P = αβ)
α² + β²	S² − 2P
(α − β)²	S² − 4P
\|α − β\|	√(S² − 4P)
α³ + β³	S³ − 3PS = S(S² − 3P)
α³ − β³	(α−β)(α²+αβ+β²) = (α−β)(S²−P)
α²β + αβ²	P · S
1/α + 1/β	S / P
α² + β² + γ²	(α+β+γ)² − 2(αβ+βγ+γα)

✏️ Quadratic — Worked Examples

Find zeroes of x² − 3x − 10
Factorise: (x−5)(x+2) = 0 → zeroes: 5 and −2
Verify: S = 5+(−2) = 3 = 3/1 ✓ P = 5×(−2) = −10 = −10/1 ✓

Form quadratic with zeroes 2+√3 and 2−√3
Sum = 4 Product = (2+√3)(2−√3) = 4−3 = 1
p(x) = x² − 4x + 1

If α+β = 5, αβ = 6 → find α²+β² and 1/α+1/β
α²+β² = 25 − 12 = 13 1/α+1/β = 5/6

Reciprocal zeroes: if αβ = 1
If zeroes are α and 1/α, then αβ = 1 = c/a ⟹ c = a.
e.g. 5x²+13x+5 → c = a = 5 ✓

✏️ Cubic — Worked Example

p(x) = x³ − 4x² + 5x − 2, one zero = 1
α = 1, so α+β+γ = 4 → β+γ = 3
αβγ = 2 → βγ = 2
Quadratic for β, γ: t² − 3t + 2 = 0 → (t−1)(t−2) = 0
All zeroes: 1, 1, 2

Alternate: use Division
Divide p(x) by (x−1) → quotient = x²−3x+2
Solve x²−3x+2 = (x−1)(x−2) = 0 → zeroes: 1, 1, 2

Zeroes in AP: a−d, a, a+d
Sum = 3a = −b/a → find a first.
Sum of pairs = 3a² − d² = c/a → find d.

🔄 Methods Comparison

Task	Best Method	Steps / Formula
Find zeroes of quadratic	Factorisation	Split middle term; equate each factor to 0
Verify zeroes of quadratic	Vieta's Formulas	Check α+β = −b/a AND αβ = c/a
Form quadratic from zeroes	Sum & Product	x² − (Sum)x + Product
Find all zeroes of cubic	Divide by known factor	Divide p(x) by (x−α) → quadratic → solve
Divide polynomials	Polynomial Long Division	p(x) = g(x)·q(x) + r(x); ensure deg r < deg g
Find k when remainder given	Long division + compare	Divide and equate remainder coefficients

🧠 Memory Tricks & Mnemonics

Quadratic signs: "Minus, Plus"

α+β = −b/a (minus sign before b). αβ = +c/a (plus, no sign change). Never mix them up!

Cubic signs: "−, +, −"

α+β+γ = −b/a, αβ+βγ+γα = +c/a, αβγ = −d/a. Signs alternate starting with minus. Think: −b, +c, −d.

Form Quadratic: "minus Sum, plus Product"

x² − (Sum)x + (Product). The sum gets a minus sign; the product gets a plus sign.

α²+β² = "Square minus Twice"

(α+β)² − 2αβ. Square the sum, then subtract twice the product. Very common board question.

Division Algorithm: "DQGR"

Dividend = Divisor × Quotient + Remainder. p(x) = g(x)·q(x) + r(x). Always verify by expanding.

Graph touches → Repeated zero

If the parabola only touches the x-axis (tangent), the discriminant b²−4ac = 0 and the zero is a double root.

Finding other cubic zeroes

Given α: use sum/product to get β+γ and βγ. Solve t²−(β+γ)t+βγ = 0. OR simply divide by (x−α).

Quadratic Formula (Discriminant check)

x = [−b ± √(b²−4ac)] / 2a. D>0: 2 distinct zeroes. D=0: 1 repeated. D<0: no real zeroes.

📊 Quick Reference — All Vieta Formulas

Polynomial	Zeroes	Sum (−b/a)	Sum of Pairs (c/a)	Product
ax + b	α = −b/a	—	—	—
ax² + bx + c	α, β	α+β = −b/a	—	αβ = c/a
ax³ + bx² + cx + d	α, β, γ	α+β+γ = −b/a	αβ+βγ+γα = c/a	αβγ = −d/a

Quadratic Discriminant D = b² − 4ac. Two distinct zeros: D > 0. Equal (repeated): D = 0. No real zeros: D < 0.

Division Algorithm Check Always verify: expand g(x)·q(x)+r(x) and confirm it equals p(x). Also verify deg(r) < deg(g).