Polynomials

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial function of degree \( n \) is:

\( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

Guidance, Clarification:

• Graphs of Polynomial Functions:
  • The graph of a polynomial function of degree \( n \) will intersect the x-axis at most \( n \) times.
  • The behavior of the graph near the x-axis depends on the roots (real and complex) of the polynomial.
• Zeros, Roots, and Factors:
  • Zeros/Roots: The values of \( x \) for which \( f(x) = 0 \).
  • Factors: If \( r \) is a root of \( f(x) \), then \( (x – r) \) is a factor of the polynomial.
• Factor Theorem:
  • If \( f(r) = 0 \), then \( (x – r) \) is a factor of \( f(x) \).
• Remainder Theorem:
  • When a polynomial \( f(x) \) is divided by \( (x – r) \), the remainder is \( f(r) \).
• Sum and Product of Roots:
  • For a polynomial equation of degree \( n \): \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0 \).
  • Sum of the roots: \( -\frac{a_{n-1}}{a_n} \).
  • Product of the roots: \( (-1)^n \cdot \frac{a_0}{a_n} \).