Maths I — Week 2: Polynomials, Quadratics, and Coordinate Geometry

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Week 2: Polynomials and Quadratics

1. Anatomy of a Polynomial

A polynomial is an expression of the form

a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0

where all exponents are non-negative integers.

Degree: The highest power. The degree controls the maximum number of roots.
Leading Coefficient: $a_n$ (the coefficient of the highest power).

2. The Quadratic Equation

Standard form:

ax^2 + bx + c = 0

, where

a \neq 0

The roots are given by the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The Discriminant $\Delta = b^2 - 4ac$

The discriminant completely determines the nature of the roots before solving:

$\Delta$	Nature of Roots	Graph Behaviour
$\Delta > 0$	2 distinct real roots	Parabola crosses x-axis twice
$\Delta = 0$	1 repeated real root	Parabola touches x-axis (tangent)
$\Delta < 0$	2 complex conjugate roots	Parabola never crosses x-axis

Trap: If $\Delta < 0$ and someone asks "find the roots using the formula", the answer will involve $\sqrt{\text{negative}}$ — a complex (imaginary) number, not a real root. Always check $\Delta$ first.

3. Vieta's Formulas (Root Relationships without Finding Roots)

For a quadratic

ax^2 + bx + c = 0

with roots

\alpha

and

\beta

\alpha + \beta = -\frac{b}{a}

\alpha \cdot \beta = \frac{c}{a}

This lets you reconstruct the equation from the roots or find properties of roots without solving the full quadratic.

4. Factorization Strategies

Factor out GCD: $6x^2 + 9x = 3x(2x + 3)$ .
Product-Sum Method: Find two numbers that multiply to $ac$ and add to $b$ .
Difference of Squares: $a^2 - b^2 = (a+b)(a-b)$ .
Perfect Square Trinomial: $a^2 + 2ab + b^2 = (a+b)^2$ .

5. Flashcards

6. Problem Solving Combinations & Solutions

Type 1: Find equation given roots

Q: Find a quadratic with roots

3

and

-2

. Solution: Using Vieta's — Sum

= 1

, Product

= -6

. Equation:

x^2 - x - 6 = 0

Type 2: Find properties from equation without solving

Q: For

2x^2 - 5x + 3 = 0

, find

\alpha^2 + \beta^2

. Solution:

\alpha + \beta = 5/2

\alpha\beta = 3/2

\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = 25/4 - 3 = 13/4

Type 3: Nature of roots analysis

Q: For what values of

k

does

kx^2 + 4x + 1 = 0

have equal roots? Solution: Equal roots

\implies \Delta = 0 \implies 16 - 4k = 0 \implies k = 4

Common Traps

$a = 0$ makes it linear, not quadratic.
Repeated roots ( $\Delta=0$ ) means only ONE distinct root, but algebraically it counts as two.
Complex roots always come in conjugate pairs for real-coefficient polynomials.

7. Coordinate Geometry Toolkit (Week 2 Graded Focus)

Week 2's graded assignment heavily tests coordinate geometry alongside polynomials. Master these tools.

7.1 Key Formulas

Formula	Expression
Distance between $(x_1,y_1)$ and $(x_2,y_2)$	$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Midpoint	$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
Section Formula (Internal)	$\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)$ for ratio $m:n$
Slope of a line	$m = \frac{y_2 - y_1}{x_2 - x_1}$
Slope of perpendicular	$m_\perp = -\frac{1}{m}$
Point-slope form	$y - y_1 = m(x - x_1)$

Tip

Perpendicular Slope Rule: If the slope of a line is

3

, the perpendicular slope is

-1/3

. If slope is

-1/2

, perpendicular is

2

. Always negate and flip.

7.2 Pattern — Perpendicular Bisector

What to recognize: Given two points

A

and

B

, find the perpendicular bisector.

Abstract Solution (Strategy)

[Find midpoint]: The perpendicular bisector passes through the midpoint of $AB$ .
[Find slope of AB]: $m_{AB}$ .
[Perpendicular slope]: $m_\perp = -1/m_{AB}$ .
[Point-slope form]: Write the line through the midpoint with slope $m_\perp$ .

Worked Example:

Perpendicular bisector of segment joining $(1, 0)$ and $(5, 4)$ .

Midpoint: $(3, 2)$ .
Slope of segment: $\frac{4-0}{5-1} = 1$ .
Perpendicular slope: $-1$ .
Line: $y - 2 = -1(x - 3) \implies y = -x + 5$ .

8. Additional Flashcards

9. Practice Targets

Solve Graded Assignment 2 (Coordinate Geometry + Polynomials).
Find $\alpha^3 + \beta^3$ using Vieta's (hint: factor the sum of cubes identity).
Determine nature of roots for 5 random quadratics using discriminant only.
Prove that the diagonals of a square are perpendicular bisectors using the section and slope formulas.

10. Connections

Connects to	How
Week 1 — Functions	Quadratics are the first non-linear functions we analyze.
Week 3 — Limits	Limit of a quadratic at a removable discontinuity is resolved by factoring (the same quadratic methods).
Statistics I	The normal distribution curve is bell-shaped — a Gaussian, which is related to $e^{-x^2}$ .