Polynomials and Quadratic Functions: Precalculus Study Notes

Polynomials

Definition and Structure

A polynomial is a function expressed as a sum of powers of a variable, each multiplied by a coefficient. The general form is:

• General Form: $a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$

• Coefficients: $a_i$ are real numbers

• Degree: $n$ (the highest power of $x$ with a nonzero coefficient)

• Leading Coefficient: $a_n$ (coefficient of the highest degree term)

• Constant Term: $a_0$ (term without $x$)

Example: $4x^3 + 3x^2 + 5x + 9$ is a polynomial of degree 3 with leading coefficient 4 and constant term 9.

Domain and Range

• Domain: All real numbers ($\mathbb{R}$)

• Range: Depends on the degree and leading coefficient

• Even Degree: As $x \to \pm\infty$, $f(x) \to \infty$ or $f(x) \to -\infty$ depending on the sign of $a_n$

• Odd Degree: As $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to -\infty$ (or vice versa)

Types of Polynomials

• Linear: Degree 1 ($ax + b$)

• Quadratic: Degree 2 ($ax^2 + bx + c$)

• Higher Degree: Cubic ($ax^3 + bx^2 + cx + d$), Quartic ($ax^4 + \ldots$), etc.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states:

• A polynomial of degree $n$ has at most $n$ roots (solutions), counting multiplicity.

• Over the complex numbers, every polynomial of degree $n$ has exactly $n$ roots.

Example: $x^2 - x - 6$ has two roots: $x = 3$ and $x = -2$.

Roots and Multiplicity

• Root: A value $r$ such that $f(r) = 0$

• Multiplicity: The number of times a root is repeated

• Factored Form: $a(x - r_1)(x - r_2)\ldots(x - r_n)$

Linear Functions

Definition and Properties

A linear function is a degree 1 polynomial:

• General Form: $f(x) = a_1 x + a_0$ or $y = mx + b$

• Constant Rate of Change: The slope $m$ is constant

• Y-intercept: $b$ (where the graph crosses the y-axis)

• X-intercept: $x$ value where $y = 0$; $x = -\frac{b}{m}$

Solving Linear Equations

• Move all $x$ terms to one side and constants to the other

• Solve for $x$

Example: Solve $4(x - 1) = 9$

• Expand: $4x - 4 = 9$

• Add 4: $4x = 13$

• Divide by 4: $x = \frac{13}{4}$

Point-Slope Form

• Formula: $y - y_0 = m(x - x_0)$

• Used to write the equation of a line given a point $(x_0, y_0)$ and slope $m$

Quadratic Functions

Definition and Graphs

A quadratic function is a degree 2 polynomial:

• General Form: $y = ax^2 + bx + c$, $a \neq 0$

• Graph: Parabola

• Vertex: The highest or lowest point of the parabola

• Axis of Symmetry: Vertical line through the vertex

• Concavity: If $a > 0$, parabola opens upward (concave up); if $a < 0$, opens downward (concave down)

Vertex Form

• Vertex Form: $f(x) = a(x - p)^2 + q$

• Vertex at $(p, q)$

• Allows easy sketching and identification of vertex

Example: $y = 2(x - 3)^2 + 5$ has vertex at $(3, 5)$ and opens upward.

Finding Vertex Form

• Complete the Square: Rewrite $ax^2 + bx + c$ in vertex form by completing the square

• Evaluation Method: Use formulas to find vertex

Vertex Formula:

• $x = -\frac{b}{2a}$

• $y = f\left(-\frac{b}{2a}\right)$

Example: For $y = x^2 + 4x + 2$:

• Vertex $x$-coordinate: $x = -\frac{4}{2 \times 1} = -2$

• Vertex $y$-coordinate: $y = (-2)^2 + 4(-2) + 2 = 4 - 8 + 2 = -2$

• Vertex: $(-2, -2)$

Solving Quadratic Equations

There are three main methods to solve quadratic equations:

• Factoring: Express as product of binomials and set each factor to zero

• Completing the Square: Rewrite in the form $(x - p)^2 = q$ and solve for $x$

• Quadratic Formula: For $ax^2 + bx + c = 0$:

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

Discriminant and Types of Solutions

• Discriminant: $D = b^2 - 4ac$

• If $D > 0$: Two distinct real solutions

• If $D = 0$: One real solution (double root)

• If $D < 0$: Two complex solutions (no real roots)

Discriminant ($D$)	Number of Real Solutions	Type of Roots
$D > 0$	2	Distinct real roots
$D = 0$	1	Repeated real root
$D < 0$	0	Complex roots

Example: Solve $x^2 - 4x + 3 = 0$

• $a = 1$, $b = -4$, $c = 3$

• $D = (-4)^2 - 4 \times 1 \times 3 = 16 - 12 = 4$

• Two real solutions: $x = \frac{4 \pm 2}{2} = 3, 1$

Additional info: Some formulas and examples were expanded for clarity and completeness.