MTH101 Lecture 4: Variables and Constants

Course Information

• Course: MTH101
• Institution: Batterjee Medical College
• Instructor: Ms. Afaf Alqahtani

Lecture 4: Variables and Constants

• Variables are literal numbers that can have different values
• Constants have a fixed value
• In algebra, constants are typically denoted by letters (a, b, c) and variables by letters (x, y, z).
• All real numbers are considered constants
• Examples of constants: 6, -3.5, √15, 11
• Examples of variables: 1/2x, 9r, −2y, 4x + y
• Variables' values depend on the values of any unknown variables.

Lecture 4: Algebraic Expressions and Polynomials

• An algebraic expression is a combination of numbers, variables, and arithmetic operations (+, -, ×, ÷).
• Separate parts of an algebraic expression are called terms.
• The first term's sign is often omitted in algebraic expressions.
• Example: 2x - 5y + 6
• Terms in this example are 2x, -5y and 6.

Lecture 4: Coefficients

• Coefficients are the numerical factors in front of the variables in an algebraic expression.
• In an expression like 2x, the coefficient of x is 2
• In 2/3y, the coefficient of y is 2/3
• If no number is written with a variable, the coefficient is 1 (e.g., the coefficient of x is 1).

Lecture 4: Polynomials

• A polynomial is an algebraic expression where:
  • No variables are in the denominator
  • Exponents of the variables are whole numbers (0, 1, 2, ...)
  • All coefficients are real numbers.
• Examples of Polynomials: x - 3y, -3x² + 2x – 1, 2x⁴y³ – 9x³y² + 6xy + 8
• Examples that are NOT Polynomials: 2/x, x⁻² + 4, √x − 5y + z

Lecture 5: Degree of Terms and Polynomials

• The degree of a term is the sum of the exponents of its variables.
• Non-zero constants have degree 0.
• Examples:
  • In 2x²y², the degree is 2 + 2 = 4
  • In 3x²y³, the degree is 2 + 3 = 5
  • In 8xy, the degree is 1 + 1 = 2
  • In 2 (the constant), the degree is 0.
• The degree of a polynomial is the highest degree of any term in the polynomial.

Lecture 5: Classifying Polynomials

• Polynomials are classified by degree:
  • Degree 0: Constant polynomials (e.g., 3)
  • Degree 1: Linear polynomials (e.g., 5x + 4)
  • Degree 2: Quadratic polynomials (e.g., 3x² - 5x + 4)
  • Degree 3: Cubic polynomials (e.g, 2x³ − x² + x −3)
• Polynomials are also classified by the number of terms:
  • Monomial: One term (e.g., 5x, 2x², -3x⁶, 8)
  • Binomial: Two terms (e.g., x - 1, 3y - 1/2)
  • Trinomial: Three terms (e.g., x + y + 2, 3x² − 5x + 4 )

Lecture 5: Evaluation of a Polynomial

• Evaluate a Polynomial by substituting the given value of the variable into the polynomial and then computing the result.
• Examples: Evaluates P(x) = 3x² +x-5, where x = 2 is calculated by substituting x=2 to result to 9

Lecture 5: Zero of a Polynomial

• The zero of a polynomial is a value of the variable that makes the polynomial equal to zero.
• Example: P(x) = x² -3x +2
  • P(1) = (1)² -3(1) +2 = 0
  • P(2) = (2)² -3(2) +2 = 0 then x=1, x=2 are the zeroes.

Lecture 6: Linear Equations in One Variable

• A linear equation in one variable has the form ax + b = c, where a, b, and c are constants and x is the variable.
• To solve linear equations, use the addition/subtraction and multiplication/division rules.
• These rules state that performing the same operation on both sides of an equation does not change the solution(s). For Example:
  • 2x - 3 = 13 // Add 3 to both sides
  • 2x = 16 // Divide both sides by 2
  • x = 8

Lecture 6: Consecutive Integers

• Consecutive integers are integers that follow each other in order (e.g., 1, 2, 3, etc).
• Consecutive even/odd integers follow the same order, but only even or only odd numbers are used.
• The difference between any two consecutive integers is 1 and consecutive even/odd integers is 2
• Relationships among consecutive integers (even/odd) allows solving problems with one variable and one equation.

Lecture 7: Quadratic Equations in One Variable

• A quadratic equation in one variable has the form ax² + bx + c = 0, where a, b, and c are constants and x is the variable.
• Quadratic equations may have 0, 1, or 2 real solutions.
• If 'a' is zero, it is no longer a quadratic equation and becomes linear instead.
• The quadratic formula is used to solve quadratic equations
  • x = (-b ± √(b² - 4ac)) / 2a

Exercises

• Provided exercises for each topic reinforce the material presented in class.