Welcome to Polynomials!

Today, we're embarking on a journey to understand polynomials. Polynomials are fundamental in algebra, and mastering the vocabulary is the first step to success. We'll cover degrees, terms, coefficients, and how to put these expressions in their proper order. Get ready to boost your math skills!

Key Vocabulary

  • Monomial: A number, a variable, or a product of numbers and variables with whole-number exponents. Examples: $5$, $x$, $-7xy$, $0.5x^4$
  • Polynomial: A monomial or the sum or difference of monomials.
  • Term: Parts of an expression that are being added or subtracted.
  • Coefficient: The numerical factor of a term. For example, in the term $4x^3$, the coefficient is $4$.
  • Degree of a Monomial: The sum of the exponents of the variables in the monomial. A constant has a degree of 0. For example, the degree of $4p^4q^3$ is $4+3=7$.
  • Degree of a Polynomial: The highest degree of any term in the polynomial.
  • Standard Form of a Polynomial: A polynomial with one variable written with the terms in order from greatest degree to least degree.
  • Leading Coefficient: When a polynomial is in standard form, the coefficient of the first term.

Examples

Finding the Degree of a Monomial

Let's find the degree of a few monomials:

  • $4p^4q^3$: The degree is $4 + 3 = 7$.
  • $7ed$: The degree is $1 + 1 = 2$.
  • $3$: The degree is $0$.

Finding the Degree of a Polynomial

Now, let's find the degree of some polynomials:

  • $11x^7 + 3x^3$: The degree is $7$ (the highest degree of the terms).
  • $\frac{1}{3}w^2z + \frac{1}{2}z^4 - 5$: The degree is $4$ (the highest degree of the terms).

Writing Polynomials in Standard Form

Let's write a polynomial in standard form and identify the leading coefficient:

Given: $6x - 7x^5 + 4x^2 + 9$

Standard Form: $-7x^5 + 4x^2 + 6x + 9$

Leading Coefficient: $-7$

Classifying Polynomials

Polynomials can be classified by their degree and number of terms:

  • Degree 0: Constant (e.g., $6$)
  • Degree 1: Linear (e.g., $-2x$)
  • Degree 2: Quadratic (e.g., $18x^2 - 12x + 5$)
  • Degree 3: Cubic (e.g., $5n^3 + 4n$)
  • 1 Term: Monomial
  • 2 Terms: Binomial
  • 3 Terms: Trinomial
  • 4 or More Terms: Polynomial

Homework

Complete the worksheet to practice these concepts. Remember, understanding these fundamentals will set you up for success in more advanced algebra topics. Keep practicing, and don't hesitate to ask questions!