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Week 2: Polynomials and Quadratic Equations

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Week 2: Polynomials and Quadratic Equations

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Course: Jan 2026 - Mathematics I Difficulty: Intermediate Focus: Algebraic expressions, degrees, roots, and graphing basics.

1. Introduction to Polynomials

A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

General form of a polynomial

P(x)

in one variable

x

P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0

Where:

$a_n, a_{n-1}, \dots, a_0$ are constants (coefficients).
$a_n \neq 0$ implies the polynomial has Degree $n$ .
$x$ is the variable.

1.1 Types by Degree

Degree 0: Constant Polynomial, e.g., $P(x) = 5$ .
Degree 1: Linear Polynomial, e.g., $P(x) = ax + b$ . Graph is a straight line.
Degree 2: Quadratic Polynomial, e.g., $P(x) = ax^2 + bx + c$ . Graph is a parabola.
Degree 3: Cubic Polynomial, e.g., $P(x) = ax^3 + bx^2 + cx + d$ .

2. Roots (Zeroes) of a Polynomial

A real number

k

is said to be a zero (or root) of a polynomial

P(x)

P(k) = 0

. Geometrically, the zeroes of a polynomial are the

x

-coordinates of the points where its graph intersects the

x

-axis. A polynomial of degree

n

can have at most

n

real roots.

2.1 Remainder and Factor Theorems

Remainder Theorem: If a polynomial $P(x)$ is divided by a linear binomial $(x - a)$ , the remainder is $P(a)$ .
Factor Theorem: A polynomial $P(x)$ has a factor $(x - a)$ if and only if $P(a) = 0$ .

3. Quadratic Equations

A quadratic equation is a polynomial equation of degree 2. Standard form:

ax^2 + bx + c = 0 \quad (a \neq 0)

3.1 Methods of Solving

Factoring: Finding two binomials that multiply to the quadratic expression.
Completing the Square: Manipulating the equation to form a perfect square trinomial.
Quadratic Formula: Derived from completing the square. The roots are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

3.2 The Discriminant

The expression under the square root in the quadratic formula,

\Delta = b^2 - 4ac

, is called the Discriminant. It determines the nature of the roots:

If $\Delta > 0$ : Two distinct real roots. The parabola crosses the $x$ -axis twice.
If $\Delta = 0$ : One repeated real root. The parabola touches the $x$ -axis at its vertex.
If $\Delta < 0$ : No real roots (two complex conjugate roots). The parabola does not intersect the $x$ -axis.

3.3 Sum and Product of Roots

Let

\alpha

and

\beta

be the roots of

ax^2 + bx + c = 0

Sum of Roots ( $\alpha + \beta$ ): $-\frac{b}{a}$
Product of Roots ( $\alpha \cdot \beta$ ): $\frac{c}{a}$

4. Graphing Parabolas

The graph of

y = ax^2 + bx + c

is a parabola.

Opening: Opens upwards if $a > 0$ , downwards if $a < 0$ .
Vertex: The peak or lowest point. The $x$ -coordinate of the vertex is $h = -\frac{b}{2a}$ . The $y$ -coordinate is $k = f(h)$ . It can also be written in vertex form: $y = a(x-h)^2 + k$ .
Axis of Symmetry: The vertical line passing through the vertex, $x = -\frac{b}{2a}$ .
$y$ -intercept: Occurs at $x=0$ , which is the point $(0, c)$ .

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