Prime Factorization and Polynomial Operations

Prime Numbers and Prime Factorization

• The uniquely even prime number is $2$.
• The number $1$ is not considered a prime number.
• The sequence of the smallest prime numbers includes: $2$, $3$, $5$, $7$, $13$, $17$, $19$, and $23$.
• Knowledge of prime numbers is essential for the process of factoring.
• The goal of prime factorization is to factor a given number such that every factor produced is a prime number.
• Example: Factoring the number $100$.
  • Starting with division by the smallest prime: $100 \div 2 = 50$.
  • The next step involves factoring $50$, which equals $2 \times 25$.
  • The number $25$ is further factored into $5 \times 5$.
  • The complete prime factorization of $100$ is expressed as a series of prime factors. This process is specifically identified as the prime factor of addition.

Polynomial Classifications and Terminology

• Polynomials may involve multiple variables, including $x$ and $y$.
• The general form of a term in a polynomial can be represented as $Ax^k$.
• Polynomials are classified based on the number of terms they contain:
  • Monomial: A polynomial containing exactly one term.
  • Binomial: A polynomial containing exactly two terms. For example, an expression where $2x$ is the first term.
  • Trinomial: A polynomial containing exactly three terms.
  • For polynomials with more than three terms, they are typically referred to by the specific number of terms present.
• The Constant Zero:
  • The number zero is represented as $0.0$.
  • This is described as a particular polynomial to which no degree is assigned.

Operations and Special Products

• There are four primary operations used most frequently with polynomials:
  • Adding
  • Subtracting
  • Multiplying
  • Dividing
• These algorithms are employed to determine the sum, difference, product, and quotient of two or more polynomials.
• Special product formulas are repetitive in nature and are required memorization for efficient calculation.

Factoring and Like Terms

• Factoring is regarded as perhaps the most important skill when working with polynomials.
• The process of factoring involves identifying common factors within the terms of an expression.
• Common factors are those factors present in every term.
• To perform factoring:
  • Identify and write the common factors separately.
  • Place the remaining parts of the terms within parentheses.
  • Perform necessary numerical operations, such as $-3 + 5$.
• Resulting expressions, such as $2xy^2$, are considered much simpler than their non-factored counterparts.
• Combining Like Terms:
  • Like terms are defined as terms that share the exact same variable and the exact same power (degree).
  • Example: Given $7x^4$ and $-3x^4$, the common character is $x^4$. By calculating $7 - 3$, the term simplifies to $4x^4$.
  • Example: The term $6x^3$ is the result of adding $6x^3$ to zero or another like term.
  • Example: Handling negative signs is critical, such as expressions involving $-x^2$ minus another value.

Questions & Discussion

• Is one a prime number?
  • No, there is no change to that rule; one is not a prime number. The smallest prime number is $2$.
• How many operations are used most often with polynomials?
  • There are four operations: adding, subtracting, multiplying, and dividing.
• What is the structure of the review?
  • The review consists of $50$ total items, many of which are multiple-choice, covering approximately $12$ core concepts. A website is available for temporary use to facilitate this review.