Algebraic Expressions and Polynomials

Polynomials

Basic Concepts

• Terms: In an algebraic expression, terms are the individual components separated by addition or subtraction. For example, in the expression $3x^2 - x + 5$, the terms are $3x^2$, $-x$, and $5$.
• Term Definition: A term can be a constant, a variable, or a product of a constant and a variable.
• Numerical Coefficient: The constant factor in a term. For example, in the term $3x^2$, the numerical coefficient is $3$. In the term $-x$, the numerical coefficient is $-1$.
• Literal Coefficient: The variable part of a term, including its exponent. For example, in the term $3x^2$, the literal coefficient is $x^2$. In the term $-x$, the literal coefficient is $x$.
• Coefficient: When the term 'coefficient' is used alone, it refers to the numerical coefficient.
• Constant: A term without a variable. It's usually referred to as the term without a variable. In the expression $3x^2 - x + 5$, the constant term is $5$.
• Base: In a literal coefficient like $x^2$, $x$ is the base.
• Exponent: In a literal coefficient like $x^2$, $2$ is the exponent.
• Degree of a Term: The highest exponent or the highest sum of the exponents of the variables in a term.
  • Example 1: In $3x^2 - x + 5$, the degree is $2$. The degree of term $3x^2$ is 2, the degree of term $-x$ is 1, and the degree of term $5$ is 0.
  • Example 2: In $3x^2y^3 - x^4y^3$, the degree of the term $3x^2y^3$ is $2 + 3 = 5$, and the degree of the term $-x^4y^3$ is $4 + 3 = 7$. Therefore, the degree of the expression is $7$.
• Similar Terms: Terms that have the same literal coefficients.
  • Example 1: $3x^2$ and $-5x^2$ are similar terms because their literal coefficients are both $x^2$.
  • Example 2: $5x$ and $5x^2$ are NOT similar because their literal coefficients, $x$ and $x^2$, are not the same.
  • Example 3: $2x^3y^2$ and $-4x^2y^3$ are NOT similar because their literal coefficients, $x^3y^2$ and $x^2y^3$, are not the same.

Polynomials

• Definition: A polynomial is an algebraic expression where each term is a constant, a variable, or a product of a constant and a variable, in which the variable has a whole number (non-negative number) exponent.
• Types of Polynomials: A polynomial can be a monomial, binomial, trinomial, or a multinomial.
• Non-Polynomial Conditions: An algebraic expression is NOT a polynomial if:
  • The exponent of the variable is NOT a whole number (e.g., $x^{1/2}$, $x^{-1}$).
  • The variable is inside the radical sign (e.g., $\sqrt{x}$).
  • The variable is in the denominator (e.g., $\frac{1}{x}$).

Kinds of Polynomials According to the Number of Terms

• Monomial: A polynomial with only one term (e.g., $5x^2$).
• Binomial: A polynomial with two terms (e.g., $x + 3$).
• Trinomial: A polynomial with three terms (e.g., $x^2 - 2x + 1$).
• Polynomial: A polynomial with four or more terms (also sometimes referred to as a multinomial).

Kinds of Polynomials According to Degree

• Constant: A polynomial of degree zero (e.g., $7$).
• Linear: A polynomial of degree one (e.g., $2x + 1$).
• Quadratic: A polynomial of degree two (e.g., $3x^2 - x + 2$).
• Cubic: A polynomial of degree three (e.g., $x^3 + 2x^2 - x + 5$).
• Quartic: A polynomial of degree four (e.g., $x^4 - 3x^2 + 1$).
• Quintic: A polynomial of degree five (e.g., $2x^5 + x^3 - x + 8$).
• Polynomials of degree six or higher are typically referred to as