Polynomials (Conceptual Notes)

What is a Polynomial?

• A polynomial is an algebraic expression made of one or more terms.

• Each term is made up of a coefficient (number) and a variable (letter), sometimes raised to a power.

• Polynomials are used in algebra, graphs, and modeling real-life situations.

Types of Polynomials (Based on Number of Terms)

1. Monomial → 1 term

2. Binomial → 2 terms

3. Trinomial → 3 terms

4. Polynomial with more than 3 terms → simply called a polynomial

Degree of a Polynomial

• The degree is the highest power of the variable in the polynomial.

• Helps in classifying polynomials and understanding their behavior in graphs.

Operations with Polynomials

• Addition/Subtraction: Combine like terms (terms with same variable and exponent).

• Multiplication: Each term of one polynomial multiplies every term of the other polynomial.

• Factorization: Breaking a polynomial into simpler expressions that multiply to give the original polynomial.

• Zeros of a Polynomial: Values of the variable that make the polynomial zero; useful for graphing.

Key Points for Exams

• Polynomial = sum of terms with numbers and variables

• Types: Monomial, Binomial, Trinomial

• Degree = highest exponent

• Combine like terms for addition/subtraction

• Factorization = split into simpler multiplying expressions

• Zeros = values that make polynomial zero

2. Linear Equations in Two Variables (Conceptual Notes)

What is a Linear Equation in Two Variables?

• An equation involving two unknowns (x and y) that forms a straight line on a graph.

• Every solution of the equation is a pair of numbers that satisfies it.

Solutions of a Linear Equation

• A solution is a pair of values that makes the equation true.

• There are infinite solutions because many pairs can satisfy the equation.

• Each solution can be represented as a point on the coordinate plane.

Graphical Representation

• Plot the solutions as points and draw a straight line through them.

• The line represents all possible solutions of the equation.

Types of Solutions for Two Equations

1. Unique Solution: Lines intersect at one point.

2. Infinite Solutions: Lines coincide (overlap completely).

3. No Solution: Lines are parallel and never meet.

Key Points for Exams

• Linear equation in two variables → represents a straight line

• Each solution = ordered pair (x, y)

• Infinite solutions → points on the line

• Graph is the visual representation of solutions

• Two equations → solution can be one, infinite, or none