Identify and Evaluate Polynomials

Learning Outcomes

  • Understand what a polynomial is and how to evaluate it for specific values.

Definition of Polynomial

  • A polynomial is an expression that includes:

    • A sum or difference of terms

    • Each term can consist of:

      • Real numbers

      • Variables

      • Products of real numbers and variables with non-negative integer exponents

  • Non-negative integers include: 0, 1, 2, 3, ...

Identifying Polynomials

Criteria for a Polynomial

  • Use the table below to distinguish between polynomials and non-polynomials:

    IS a Polynomial

    IS NOT a Polynomial

    2x² − 1

    2/x²

    2y² + 4

    x² + √2

    2x² - 12x - 9

    2x^(1/2) + 4

    y⁴ − y³

    a+7

Characteristics of Polynomials

  • Polynomials cannot:

    • Have variables in the denominator

    • Have roots equivalent to rational exponents; they must have integer exponents

Types of Polynomials

Basic Components

  • Monomial: A single term (e.g., 3y)

  • Binomial: A polynomial with two terms (e.g., 2x − 9)

  • Trinomial: A polynomial with three terms (e.g., −3x² + 8x − 7)

Degree of a Polynomial

  • Degree: The highest power of the variable in a polynomial

    • The term with the highest exponent is known as the leading term

    • The coefficient of the leading term is called the leading coefficient

  • A polynomial is in standard form when its terms are arranged in decreasing order of power.

Identifying Polynomial Expressions

Examples

  • Determine which of the following expressions are polynomials:

    1. -112x³ + 5 + 2x² ✅

    2. 5x¹² − 2x³ + 7x ✅

    3. 3 + 5 + 2 ✅

    4. 2/x ❌ (invalid since x is in the denominator)

    • Non-polynomial examples are indicated: e.g., expressions with non-integer exponents or roots.

Evaluating Polynomials

  • To evaluate a polynomial for given values:

    • Substitute the value of the variable into the polynomial

    • Apply the order of operations to compute the result

Example of Evaluation

  • Evaluate 3x² − 2x + 1 for x = -1:

    1. Substitute -1 for x:

      • 3(-1)² - 2(-1) + 1

    2. Calculate:

      • 3(1) + 2 + 1 = 3 + 2 + 1 = 6

Another Example

  • Evaluate -23p⁴ + 2p³ − p for p = 3:

    • Substitute 3:

      • -23(3)⁴ + 2(3)³ - 3

    • Evaluate:

      • Calculate powers first, then multiply and sum results.