Polynomials, Quadratic Equations, and Complex Numbers: Precalculus Study Guide

Polynomial Functions

Quadratic Equations

Quadratic equations are polynomials of degree 2, typically written in the form ax2 + bx + c = 0. Solving these equations is fundamental in algebra and precalculus.

• Standard Form: $ax^2 + bx + c = 0$

• Strategy: Move all terms to one side so the equation equals zero.

• Quadratic Formula: The roots of the quadratic equation are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Factoring: If possible, factor the quadratic into two binomials and solve for x.

• Discriminant: $D = b^2 - 4ac$ determines the nature of the roots.

Example:

• Given $x^2 - 6x = 0$, factor as $x(x - 6) = 0$ so $x = 0$ or $x = 6$.

• Given $x^2 - x - 6 = 0$, factor as $(x - 3)(x + 2) = 0$ so $x = 3$ or $x = -2$.

• Using the quadratic formula for $x^2 - 4x + 13 = 0$: $x = \frac{4 \pm \sqrt{16 - 52}}{2} = \frac{4 \pm \sqrt{-36}}{2} = \frac{4 \pm 6i}{2} = 2 \pm 3i$

Discriminant and Types of Roots

The discriminant $D = b^2 - 4ac$ is used to classify the roots of a quadratic equation:

• D > 0: Two distinct real solutions

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

• D < 0: No real solutions; two complex conjugate solutions

Example:

• For $x^2 - 4x + 13 = 0$, $D = (-4)^2 - 4 \cdot 1 \cdot 13 = 16 - 52 = -36$ (complex roots)

• For $x^2 - 4x + 4 = 0$, $D = (-4)^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0$ (one real root)

Complex Numbers

Definition and Properties

Complex numbers are numbers of the form a + bi, where a is the real part and b is the imaginary part. The imaginary unit i is defined as $i = \sqrt{-1}$.

• Standard Form: $a + bi$

• Purely Imaginary: When $a = 0$, e.g., $7i$

• Examples: $2 + 3i$, $4i$, $\sqrt{2}i$

Operations with Complex Numbers

• Addition/Subtraction: Add/subtract real parts and imaginary parts separately. Example: $(9 + 2i) + (1 + 3i) = 10 + 5i$ $(9 + 2i) - (1 + 3i) = 8 - i$

• Multiplication: Use distributive property and $i^2 = -1$. Example: $(4 + 3i)(2 + 7i) = 8 + 28i + 6i + 21i^2 = 8 + 34i - 21 = -13 + 34i$

• Conjugate: The conjugate of $a + bi$ is $a - bi$. Example: The conjugate of $4 + 7i$ is $4 - 7i$.

• Division: Multiply numerator and denominator by the conjugate of the denominator. Example: $\frac{1 + i}{3 + 2i}$ Multiply by conjugate: $\frac{1 + i}{3 + 2i} \cdot \frac{3 - 2i}{3 - 2i}$ $= \frac{(1 + i)(3 - 2i)}{(3 + 2i)(3 - 2i)}$ $= \frac{3 - 2i + 3i - 2i^2}{9 - 4i^2}$ $= \frac{3 + i + 2}{9 + 4}$ $= \frac{5 + i}{13}$

Factoring Polynomials

Factoring Quadratics

Factoring is a method to solve quadratic equations by expressing them as a product of two binomials.

• Find two numbers: Their product is ac and their sum is b.

• Example: $x^2 - x - 6 = 0$ Find numbers that multiply to -6 and add to -1: -3 and 2. Factor: $(x - 3)(x + 2) = 0$

• Perfect Square Trinomials: $a^2 + 2ab + b^2 = (a + b)^2$

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

• Sum/Difference of Cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$

Example:

• $x^2 - 9 = (x - 3)(x + 3)$

• $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$

Factoring with Leading Coefficient Not 1

When the leading coefficient a is not 1, use factoring by grouping or the AC method.

• Example: $6x^2 + 5x - 6 = 0$ Find two numbers that multiply to $6 \times -6 = -36$ and add to 5: 9 and -4. Rewrite: $6x^2 + 9x - 4x - 6$ Group: $(6x^2 + 9x) + (-4x - 6)$ Factor: $3x(2x + 3) -2(2x + 3)$ $(2x + 3)(3x - 2) = 0$

Summary Table: Types of Quadratic Roots

Discriminant (D)	Nature of Roots	Example
$D > 0$	Two distinct real roots	$x^2 - 5x + 6 = 0$
$D = 0$	One real root (double root)	$x^2 - 4x + 4 = 0$
$D < 0$	Two complex conjugate roots	$x^2 - 4x + 13 = 0$

Additional info:

• Some context and examples were inferred for clarity and completeness.

• Factoring methods and complex number operations are expanded for academic completeness.