Notes on Adding and Subtracting Polynomials

Combine like terms:
  1. $4a + 89a + 3$
  2. $7d^5c + 3d^{-1}$
  3. $5b^2b - 4 + 36^2 - 2$
Solve:
4. $5(2y - 1) - 4(6 - y)$
5. $-1(6x^2 + 3x - 8)$

Definition of a Polynomial: A polynomial is a monomial or the sum of two or more monomials.
    - A monomial is defined as: A number, a variable, or a product of a number and one or more variables.
    - A binomial is defined as: The sum of two monomials.
    - A trinomial is defined as: The sum of three monomials.
Degrees of Polynomials: Polynomials have specific degrees:
  - Degree 0: Constant
  - Degree 1: Linear
  - Degree 2: Quadratic
  - Degree 3: Cubic
  - Degree 4: Quartic
  - Degree 5: Quintic
  - Degree 6+: 6th degree, 7th degree, etc.
Degree Definitions:
  - The degree of a monomial is the sum of the exponents of all its variables.
  - A nonzero constant term has degree 0, and zero has no degree.
  - The degree of a polynomial is the greatest degree of any term in the polynomial.
  - Polynomials are named by their degree.

Task: Determine whether each expression is a polynomial. If it is, identify the degree and type (monomial, binomial, trinomial).
  - a. $8ab - 2c$
  - b. $-11.25$
  - c. $2x^2 + 3xy$
  - d. $9x^3 + 8x + 5x$
  - e. $2m^2 + 2mn - n^2$
Critical Thinking:
- Query: Is $4x - 2x^2 + 9$ written in standard form?
- Justification needed.

Standard Form of a Polynomial: When written in standard form, the terms are ordered from highest degree to lowest degree. The coefficient of the leading term is referred to as the leading coefficient.

Write the Polynomial in Standard Form:
- $4x + 12 + 2x^3 - 3x^2$
Identify Leading Coefficient: Should be established after arranging in standard form.
Check Part A: Write $5b - 10b^2 + 35 - 63$ in standard form and identify the leading coefficient in $5b - 10b^2 + 35 - b^3$ .

Process Overview: Adding polynomials involves combining like terms. While adding, you can use:
  - Method 1: Horizontal Method
    - Group and combine like terms (e.g., $(3x^2 + 9x + 27) + (2x^2 + 4x - 12)$ ).
  - Method 2: Vertical Method
    - Align like terms in columns and combine.
Example 3:
  - Find sums:
    - a. $(3x^2 - 4) + (x^2 - 9)$
    - b. $(8x^2) + (4x + 2x^2 - 9)$
Important Note: The result of adding or subtracting integers results in an integer, indicating closure under these operations. Thus, adding or subtracting polynomials results in a polynomial.

Method Overview: To subtract a polynomial, add its additive inverse. This involves negating each term of the polynomial being subtracted.
Example Approach: Find $(11x - 13 - 7x^3 - 8x^2) - (2x + 8x^2 + 20)$ through both horizontal and vertical methods.
Example 4 and 5:
- Vertically subtract digital and hard copy counts. Evaluate expressions like $(6x - 11) - (2x - 19)$ and others similarly.

Scenario 1: Album Sales
  - Define sales equations for hard copies $H = 9w + 53$ and digital copies $D = 13w + 126$ .
  - Formulate equation showing additional digital albums sold over hard copies: $S = D - H$ .
  - Predict values over 52 weeks.
Scenario 2: College Living
- Define total student population: $T = D + A$ where dorm residents $D = 11n + 8$ and total students $T = 17n + 23$ .
- Formulate equation for students living in apartments and predict future populations considering conditions.
Assumption Validation: Identify underlying assumptions needed to predict student numbers for 2020, such as unchanged enrollments and no commuter students.

Find sums or differences for given polynomial expressions:
  1. $(5m + 3) + (8m - 6)$
  2. $(3d^2 + 2d - 1) + (d^2 + 6d - 3)$
  3. $(4x^2 - 2x) - (x^2 + 8x)$
  4. $(4a^2 + 6a + 12) - (5a^2 + 8a - 17)$