Polynomials and Their Degrees

Study Notes

Degree of Terms and Polynomials

The degree of a term is the sum of the exponents of all variables in that term.
For the term -9x³y², the degree is 3 (3 from x and 2 from y).
The degree of a polynomial is defined as the highest degree among its terms.
In the polynomial 2x⁴y³ - 9x³y² + 6xy + 8, the highest degree term is 2x⁴y³, so the polynomial's degree is 7.
A constant term, such as 8, has a degree of 0.

Classification of Polynomials

Polynomials can be classified by degree and by the number of terms.
The classification by degree includes:
- Constant (degree 0)
- Linear (degree 1)
- Quadratic (degree 2)
- Cubic (degree 3), etc.
The classification by number of terms includes:
- Monomial (1 term)
- Binomial (2 terms)
- Trinomial (3 terms), etc.

Evaluation of Polynomials

To evaluate a polynomial P(x) at a specific value of x, substitute that value into the polynomial expression.
For instance, evaluating P(-1) can yield different values depending on the polynomial.
A zero of a polynomial P(x) is a value of x that makes P(x) equal to zero; for example, P(0) ≠ 0 indicates that 0 is not a zero.

Linear Equations in One Variable

Solving linear equations often utilizes the Addition/Subtraction Rule and the Multiplication/Division Rule.
The Addition/Subtraction Rule states that adding or subtracting the same number from both sides of an equation does not change its solution.
The Multiplication/Division Rule states that the same applies when multiplying or dividing both sides by a non-zero number.

Consecutive Integers

Consecutive integers differ by 1 (e.g., 1, 2, 3).
Consecutive even integers differ by 2 (e.g., 2, 4, 6).
Consecutive odd integers also differ by 2 (e.g., 1, 3, 5).
Expressions can be formulated to represent sums of consecutive integers to solve specific problems.

Quadratic Equations

A standard quadratic equation is represented as ax² + bx + c = 0.
The solutions can be found using the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a).
The discriminant (Δ) determines the nature of the roots: if Δ > 0, two distinct solutions; Δ = 0, one repeated solution; Δ < 0, no real solutions.
Examples illustrate how to compute solutions and evaluate the number of solutions for given quadratic equations.

Exercises and Solutions

Exercises provided practice on determining the degree of terms and polynomials, evaluating at certain values, and classifying polynomials.
Various exercises on solving linear equations and identifying consecutive integers reinforce the concepts discussed.

Self-Evaluation

Reflect on understanding variables (symbols representing numbers), constants (fixed values), and coefficients (numbers in front of variables) when evaluating and solving equations.

Description

This quiz explores the concept of degrees of terms and polynomials, focusing on identifying degrees from given polynomial expressions. You'll answer questions regarding specific terms and the classification of polynomials based on their degree. Test your understanding of polynomial degrees in a fun and engaging way!