Introduction to Polynomials and Basic Definitions

TL;DR

Polynomials are expressions built from variables, constants, and exponents, using only addition, subtraction, multiplication, and non-negative integer powers. They're fundamental in math, describing curves and relationships, and are categorized by their degree and number of terms. Understanding their basic parts helps you work with them effectively.

1. The Mental Model

Think of a polynomial as a recipe for a number where you mix ingredients (variables), spices (constants), and baking instructions (exponents). The rules are simple: you can only add, subtract, multiply, and use whole number powers for your variable. It's like building blocks for more complex mathematical ideas.

2. The Core Material

A polynomial is an expression made up of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Essentially, you won't see division by a variable, square roots of variables, or fractional/negative exponents on your variables.

Here's how we break down a polynomial:

Terms

Each part of a polynomial that's separated by a plus (+) or minus (-) sign is called a term. For example, in the polynomial $3x^2 + 2x - 5$, the terms are $3x^2$, $2x$, and $-5$.

Coefficients

The coefficient is the numerical factor in front of a variable in a term. In $3x^2$, the coefficient is $3$. In $2x$, the coefficient is $2$. A constant term like $-5$ can be thought of as having a coefficient of $-5$ multiplied by $x^0$ (since $x^0 = 1$).

Variables

The variables are the letters representing unknown values, like $x$, $y$, or $a$.

Exponents (Powers)

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The exponent is the small number written above and to the right of the variable, indicating how many times the variable is multiplied by itself. In $3x^2$, the exponent is $2$. For a term like $2x$, the exponent on $x$ is $1$ (it's usually not written).

Degree

The degree of a term is the value of its exponent on the variable. For a term with multiple variables, it's the sum of their exponents. The degree of the polynomial is the highest degree among all its terms. This is a super important characteristic.

Constant Term

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A constant term is a term that doesn't have a variable attached to it. It's just a number. In $3x^2 + 2x - 5$, the constant term is $-5$.

Types of Polynomials (by number of terms)

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• Monomial: A polynomial with only one term (e.g., $5x^3$, $-7$).
• Binomial: A polynomial with two terms (e.g., $x^2 + 4$, $3y - 8$).
• Trinomial: A polynomial with three terms (e.g., $2x^2 - 3x + 1$).
• Polynomials with more than three terms are generally just called "polynomials."

Types of Polynomials (by degree)

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• Constant Polynomial: Degree 0 (e.g., $7$, $-10$).
• Linear Polynomial: Degree 1 (e.g., $2x + 3$, $y - 5$).
• Quadratic Polynomial: Degree 2 (e.g., $x^2 - 4x + 7$).
• Cubic Polynomial: Degree 3 (e.g., $3x^3 + 2x^2 - x + 1$).
• Quartic Polynomial: Degree 4 (e.g., $x^4 - 5x^2 + 2$).

graph TD
    A["Polynomial Expression"] --> B["Terms (separated by + or -)"]
    B --> C["Coefficient (number part)"]
    B --> D["Variable (letter part)"]
    B --> E["Exponent (power of variable)"]
    A --> F["Degree of Polynomial (highest term's exponent)"]
    A --> G["Constant Term (number without variable)"]
    F --> H["0: Constant (e.g., '5')"]
    F --> I["1: Linear (e.g., '2x + 1')"]
    F --> J["2: Quadratic (e.g., 'x^2 + 3x - 2')"]
    F --> K["3: Cubic (e.g., '4x^3 - 7')"]
    B --> L["1 Term: Monomial (e.g., '5x')"]
    B --> M["2 Terms: Binomial (e.g., '2x + 3')"]
    B --> N["3 Terms: Trinomial (e.g., 'x^2 + x - 1')"]

3. Worked Example

Let's break down the polynomial: $5x^3 - 7x^2 + x - 12$

1. Is it a polynomial? Yes, all exponents are non-negative integers, and there's no division by variables or roots of variables.
2. How many terms? There are four terms: $5x^3$, $-7x^2$, $x$, and $-12$.
3. Identify coefficients:
   • For $5x^3$, the coefficient is $5$.
   • For $-7x^2$, the coefficient is $-7$.
   • For $x$, the coefficient is $1$ (remember, if you don't see it, it's 1).
   • For $-12$, it's the constant term.
4. Identify exponents and term degrees:
   • For $5x^3$, the exponent is $3$. Its term degree is $3$.
   • For $-7x^2$, the exponent is $2$. Its term degree is $2$.
   • For $x$, the exponent is $1$. Its term degree is $1$.
   • For $-12$, the exponent on a variable (like $x^0$) would be $0$. Its term degree is $0$.
5. What's the degree of the polynomial? The highest term degree is $3$. So, the polynomial's degree is $3$.
6. What's the constant term? The constant term is $-12$.
7. What type is it by number of terms? It has four terms, so it's generally just called a polynomial.
8. What type is it by degree? It has a degree of $3$, so it's a cubic polynomial.

4. Key Takeaways

• A polynomial uses only addition, subtraction, multiplication, and non-negative whole number exponents on variables.
• Terms are the individual parts of a polynomial, separated by plus or minus signs.
• The coefficient is the number multiplying the variable in a term.
• The degree of a polynomial is the highest exponent of any variable in its terms.
• Polynomials can be classified by their number of terms (monomial, binomial, trinomial) or by their degree (linear, quadratic, cubic, etc.).
• The constant term is the number in a polynomial that isn't multiplied by a variable.

Common Mistakes to Avoid:
- Don't confuse a coefficient with an exponent; they play different roles.
- Remember that a lone variable like $x$ has an implied coefficient of $1$ and an implied exponent of $1$.
- A number by itself (like $5$) is a polynomial with a degree of $0$, not undefined.
- Expressions with variables in the denominator (e.g., $1/x$) or under a root (e.g., $\sqrt{x}$) are not polynomials.

5. Now Try It

Take the following expressions and for each one:
1. State whether it's a polynomial or not. If it's not, explain why.
2. If it is a polynomial, identify its terms, coefficients, degree, and constant term. Then classify it by its number of terms and its degree.

Expressions:
a) $7x^4 - 2x^2 + 9x - 1$
b) $4x^2 + \frac{3}{x}$
c) $-5y^3 + 8$
d) $100$

Success looks like: For each polynomial, you'll have correctly identified all requested components and classifications. For non-polynomials, you'll have a clear reason why.