Introduction to Binomials and Polynomials

TL;DR

You'll learn about polynomials, which are expressions made of variables and numbers, and binomials, a specific type of polynomial. Understanding these basic building blocks is crucial for algebra because they appear everywhere. We'll cover how to recognize, classify, and combine them.

1. The Mental Model

Think of polynomials as mathematical "words" made from "letters" (variables) and "numbers" (coefficients) joined by addition, subtraction, and multiplication. Binomials are just very specific, two-part words. Once you know these words, you can start building sentences and stories.

2. The Core Material

In algebra, an expression is a combination of numbers, variables, and operation signs. A polynomial is a special type of algebraic expression. What makes a polynomial special? It can only have terms where variables are raised to non-negative integer powers (like $x^2$, $y^3$, $z^0$ which is just 1) and there are no variables in the denominator or under square roots.

Here's what an expression needs to be a polynomial:
* Variables (like $x$, $y$) raised to whole number exponents (0, 1, 2, 3...).
* Coefficients (numbers multiplying the variables) can be any real numbers (positive, negative, fractions, decimals).
* Operations are only addition, subtraction, and multiplication. Division by a variable is not allowed.

2.1. Terms and Coefficients

A polynomial is made up of terms. Each term is separated by a plus or minus sign.
In $3x^2 - 5x + 7$:
* $3x^2$ is a term. The coefficient is 3, the variable is $x$, and the exponent is 2.
* $-5x$ is a term. The coefficient is -5, the variable is $x$, and the exponent is 1 (we usually don't write $x^1$).
* $7$ is a term. This is called a constant term because it doesn't have a variable. You can think of it as $7x^0$.

2.2. Types of Polynomials by Number of Terms

We classify polynomials based on how many terms they have:
* Monomial: A polynomial with one term.
* Examples: $5x$, $-7y^3$, $12$
* Binomial: A polynomial with two terms. This is what your course title refers to!
* Examples: $x + 2$, $3y^2 - 4$, $ab + c$
* Trinomial: A polynomial with three terms.
* Examples: $x^2 + 2x - 1$, $y^3 - 5y + 10$
* Polynomial: Any expression with one or more terms (so monomials, binomials, and trinomials are all types of polynomials!).

2.3. Degree of a Polynomial

The degree of a term is the exponent of its variable (or sum of exponents if there are multiple variables).
* Degree of $3x^2$ is 2.
* Degree of $-5x$ is 1.
* Degree of $7$ is 0 (since $7 = 7x^0$).
* Degree of $4xy^3$ is $1 + 3 = 4$.

The degree of a polynomial is the highest degree of any of its terms.
* For $3x^2 - 5x + 7$, the degrees of the terms are 2, 1, 0. The highest is 2, so the polynomial's degree is 2.
* For $x+2$, the degrees are 1, 0. The highest is 1, so it's a first-degree binomial.
* For $y^3 - 5y + 10$, the highest degree is 3.

graph TD
    A["Algebraic Expression"] --> B{"Is it a Polynomial?"}
    B -- "No" --> C("Not a Polynomial (e.g., 1/x, sqrt(x), x^-2)")
    B -- "Yes" --> D["Polynomial"]

    D --> E{"Number of Terms?"}

    E -- "One Term" --> F["Monomial"]
    E -- "Two Terms" --> G["Binomial"]
    E -- "Three Terms" --> H["Trinomial"]
    E -- "Four or More Terms" --> I["Polynomial (general)"]

    D --> J{"Highest Term Degree?"}
    J --> K["Degree 0 (Constant)"]
    J --> L["Degree 1 (Linear)"]
    J --> M["Degree 2 (Quadratic)"]
    J --> N["Degree 3 (Cubic)"]
    J --> O["Higher Degrees"]

3. Worked Example

Let's break down the expression $5x^3 - 2xy + 8 - y^2$.

1. Is it a polynomial?

   • $5x^3$: Variable $x$ to a whole positive exponent 3. Valid.
   • $-2xy$: Variables $x, y$ to whole positive exponents 1, 1. Valid.
   • $8$: Constant term. Valid.
   • $-y^2$: Variable $y$ to a whole positive exponent 2. Valid.
     Yes, it's a polynomial.

2. How many terms does it have?
   The terms are $5x^3$, $-2xy$, $8$, and $-y^2$. It has 4 terms.

3. What's its special name (if any)?
   Since it has 4 terms, we just call it a polynomial (not a monomial, binomial, or trinomial).

4. What's the degree of each term?

   • $5x^3$: Degree 3.
   • $-2xy$: Degree $1+1=2$.
   • $8$: Degree 0.
   • $-y^2$: Degree 2.

5. What's the degree of the polynomial?
   The highest degree among its terms is 3. So, it's a 3rd-degree polynomial.

4. Key Takeaways

• A polynomial is an expression where variables have non-negative integer exponents.
• Terms are parts of an expression separated by addition or subtraction.
• Monomials have one term, binomials have two, and trinomials have three.
• The degree of a term is the sum of the exponents of its variables.
• The degree of a polynomial is the highest degree of any of its terms.
• Coefficients are the numerical factors multiplying the variables in a term.

• A constant term is a number without a variable, which has a degree of 0.

• Common mistake: Confusing $x^{-1}$ (which is $1/x$) or $\sqrt{x}$ (which is $x^{1/2}$) as valid polynomial terms. They aren't because their exponents aren't non-negative whole numbers.

• Common mistake: Forgetting that a single number (like 5) is a polynomial (specifically, a monomial of degree 0).
• Common mistake: Incorrectly summing up all exponents for the polynomial's degree instead of finding the highest degree of any single term.
• Common mistake: Forgetting that variables with no visible exponent (like $x$) implicitly have an exponent of 1.

5. Now Try It

Take any five algebraic expressions you find in a textbook or online (or make them up yourself!). For each one, identify if it's a polynomial. If it is, list its terms, determine if it's a monomial, binomial, trinomial, or just a general polynomial, and state its overall degree.
Success looks like you correctly identifying the type and degree for at least 4 out of 5 expressions.