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:

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

• Quadratic Formula: The roots of the quadratic equation are given by:

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

• Discriminant: determines the nature of the roots.

Example:

• Given , factor as so or .

• Given , factor as so or .

• Using the quadratic formula for :

Discriminant and Types of Roots

The discriminant 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 , (complex roots)

• For , (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 .

• Standard Form:

• Purely Imaginary: When , e.g.,

• Examples: , ,

Operations with Complex Numbers

• Addition/Subtraction: Add/subtract real parts and imaginary parts separately. Example:

• Multiplication: Use distributive property and . Example:

• Conjugate: The conjugate of is . Example: The conjugate of is .

• Division: Multiply numerator and denominator by the conjugate of the denominator. Example: Multiply by conjugate:

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: Find numbers that multiply to -6 and add to -1: -3 and 2. Factor:

• Perfect Square Trinomials:

• Difference of Squares:

• Sum/Difference of Cubes:

Example:

Factoring with Leading Coefficient Not 1

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

• Example: Find two numbers that multiply to and add to 5: 9 and -4. Rewrite: Group: Factor:

Summary Table: Types of Quadratic Roots

Additional info:

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

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