MATHEMATICS 10

Polynomial Function

A  polynomial is a special kind of algebraic expression where each term is a constant, a variable, or a product of constants and variable raised to whole number exponents. An algebraic expression is not a polynomial when there are square roots of variables, negative powers, and variables in the denominator of any fractions. Here are some examples:

5x^-3 — is not a polynomial term because the variable has a negative exponent.

3/x² — is not a polynomial term because the variable is in the denominator.

√x  — is not a polynomial term because the variable is inside a radical.

7x² — is a polynomial term because it obeys all the rules.

Here is a typical polynomial:   5x² + 8x -3
-the polynomial is arranged in descending order, and is of degree 2.
- 5 is the leading coefficient
- 5 x² is the leading term
- 5, 8, and 3 are the terms
- 3 is the constant term

The degree of a polynomial function is the largest power of x that appears in the polynomial. It can easily be identified if the terms are arranged in the descending order of powers.

Types of Polynomial Function:

• Zero Function
  - f(x) = 0
  - Degree: None
  - Leading Coefficient: None
  - Constant Term: None
• Constant Function
  - f(x) = 2
  - Degree: None
  - Leading Coefficient: None
  - Constant Term: 2
• Linear Function
   - g(x) = x - 7
  - Degree: 1
  - Leading Coefficient: 1
  - Constant Term: -7
• Quadratic Function
   - h(x) = 5x² + 9
  - Degree: 2
  - Leading Coefficient: 5
  - Constant Term: 9
• Cubic Function
   - p(x) = x³ - 2x² - 4x + 6
  - Degree: 3
  - Leading Coefficient: 1
  - Constant Term: 6
• Quartic Function
   - q(x) = 3x⁴ + 5x³ - x - 8
  - Degree: 4
  - Leading Coefficient: 3
  - Constant Term: -8
• Quintic Function
   - r(x) = 2x^5 - x⁴ - 4x
  - Degree: 5
  - Leading Coefficient: 2
  - Constant Term: 0
• nth degree Polynomial Function
  - f(x) = 7x^6 + 5x⁴ - 3
  - Degree: 6
  - Leading Coefficient: 7
  - Constant Term: -3