5.6.1: Resources and Key Concepts

Resources

Videos

• Remainder and Factor Theorems
  • Dividing Polynomials - The Remainder and Factor Theorems
• Rational Zeros Theorem (Rational Roots Theorem)
  • Real Zeros of Polynomials - Motivating the Rational Roots Theorem
  • Real Zeros of Polynomials - Proof of the Rational Roots Theorem
  • College Algebra - Polynomial Functions - Rational Roots Theorem - Example
  • Real Zeros of Polynomials - Examples Using the Rational Roots Theorem
• Fundamental Theorem of Algebra
  • Complex Zeros and the Fundamental Theorem of Algebra - Intro to the Fundamental Theorem of Algebra
  • Complex Zeros and the Fundamental Theorem of Algebra - Using the Fundamental Theorem of Algebra
• Complete Factorization Theorem
  • Complex Zeros and the Fundamental Theorem of Algebra - The Complete Factorization Theorem
• Complex Conjugate Theorem
  • Complex Zeros and the Fundamental Theorem of Algebra - The Zeros and Conjugate Zeros Theorems

Key Concepts

Theorems

• Remainder Theorem: If a polynomial \(f(x)\) of degree greater than or equal to one is divided by \(x-k\), the remainder is the value \(f(k)\).
• Factor Theorem: Let \(f(x)\) be a polynomial function of degree greater than or equal to one. \(k\) is a zero of \(f(x)\) if and only if \((x-k)\) is a factor of \(f(x)\).
• Rational Zeros Theorem (Rational Roots Theorem): Suppose \(f(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0\) is a polynomial of degree \(n \ge 1\), and \(a_0, a_1, \dots, a_n\) are integers. If \(r = \pm \frac{p}{q}\) is a rational zero of \(f\), in lowest terms, then \(p\) is a factor of the constant term \(a_0\), and \(q\) is a factor of the leading coefficient \(a_n\).
• Fundamental Theorem of Algebra: If \(f(x)\) is a polynomial of degree \(n \ge 1\) with complex-valued coefficients, then \(f(x)\) has at least one complex-valued zero.
• Complete Factorization Theorem (Corollary to Fundamental Theorem of Algebra): If \(f(x)\) is a polynomial of degree \(n \ge 1\) with complex-valued coefficients, then \(f(x)\) has exactly \(n\) linear factors over the complex number system. That is, \(f(x) = a(x-z_1)(x-z_2)\dots(x-z_n)\), where \(z_1, z_2, \dots, z_n\) are complex numbers (the zeros) and \(a \in \mathbb{R}\). Therefore, \(f(x)\) has \(n\) complex-valued zeros if we allow for multiplicities.
• Conjugate Zeros Theorem (Complex Conjugate Theorem): If the polynomial function \(f\) has real coefficients and a complex zero of the form \(a+bi\), then the complex conjugate of the zero, \(a-bi\), is also a zero of \(f\).

Common Mistakes

• Misapplying the Rational Zeros Theorem: Attempting to use the Rational Zeros Theorem when the polynomial coefficients are not integers.
• Stopping After Finding One Zero: When a zero is found using the Rational Zeros Theorem and synthetic division, forgetting to use the quotient (depressed polynomial) to find the remaining zeros.
• Not Accounting for Multiplicity of Zeros: When factoring or listing zeros, not indicating the multiplicity if a zero is repeated.
• Forgetting Complex Conjugate Zeros: If a polynomial has real coefficients and a complex zero \(a+bi\) is found or given, forgetting that its conjugate \(a-bi\) must also be a zero.
• Incorrectly Constructing a Polynomial from Zeros: Forgetting the leading coefficient \(a\) when constructing a polynomial from its zeros.