Podcast
Questions and Answers
What is a Local Maximum?
What is a Local Maximum?
- The graph of every polynomial function of degree n
- The y-coordinate of a turning point when the point is higher than all nearby points (correct)
- The equation f(x) = 0 has at least one solution
- The y-coordinate of a turning point when the point is lower than all nearby points
What is a Local Minimum?
What is a Local Minimum?
- The graph of every polynomial function of degree n
- The equation f(x) = 0 has at least one solution
- The y-coordinate of a turning point when the point is lower than all nearby points (correct)
- The y-coordinate of a turning point when the point is higher than all nearby points
What is the turning point of polynomial functions?
What is the turning point of polynomial functions?
The graph of every polynomial function of degree n has at most n-1 turning points.
What does the Fundamental Theorem of Algebra state?
What does the Fundamental Theorem of Algebra state?
What does the Fundamental Theorem of Algebra: Corollary specify?
What does the Fundamental Theorem of Algebra: Corollary specify?
What is the Complex Conjugate Theorem?
What is the Complex Conjugate Theorem?
What does the Irrational Conjugates Theorem state?
What does the Irrational Conjugates Theorem state?
What does Descartes' Rule of Signs indicate?
What does Descartes' Rule of Signs indicate?
What is the definition of a zero of a function?
What is the definition of a zero of a function?
What does The National Zero Theorem state?
What does The National Zero Theorem state?
Give examples of rational numbers.
Give examples of rational numbers.
What are Real Numbers?
What are Real Numbers?
What does the Factor Theorem state?
What does the Factor Theorem state?
What is the Remainder Theorem?
What is the Remainder Theorem?
What is Synthetic Division?
What is Synthetic Division?
What is Long Division in the context of polynomials?
What is Long Division in the context of polynomials?
What is Factoring by Grouping?
What is Factoring by Grouping?
What are the steps involved in factoring cubes?
What are the steps involved in factoring cubes?
What are the steps in factoring a polynomial?
What are the steps in factoring a polynomial?
What is the Difference of Two Cubes factoring pattern?
What is the Difference of Two Cubes factoring pattern?
What is the Sum of Two Cubes factoring pattern?
What is the Sum of Two Cubes factoring pattern?
What does it mean for a polynomial to be factored completely?
What does it mean for a polynomial to be factored completely?
What is a common monomial factor?
What is a common monomial factor?
What is the Difference of Two Squares pattern?
What is the Difference of Two Squares pattern?
What is a Perfect Square Trinomial?
What is a Perfect Square Trinomial?
What is a General Trinomial?
What is a General Trinomial?
What is a Common Polynomial Function of Degree 2?
What is a Common Polynomial Function of Degree 2?
What is a Common Polynomial Function of Degree 1?
What is a Common Polynomial Function of Degree 1?
Study Notes
Polynomial Functions: Key Concepts
- Local Maximum: The y-coordinate of a turning point that is higher than surrounding points.
- Local Minimum: The y-coordinate of a turning point that is lower than surrounding points.
- Turning Points: A polynomial function of degree n has at most n-1 turning points, and if it has n distinct real zeros, it has exactly n-1 turning points.
The Fundamental Theorem of Algebra
- Theorem: A polynomial of degree n (where n > 0) has at least one solution in the complex numbers.
- Corollary: A polynomial of degree n has exactly n solutions, counting multiplicity.
Zero Theorems
- Complex Conjugate Theorem: If a polynomial with real coefficients has an imaginary zero a + bi, then a - bi is also a zero.
- Irrational Conjugates Theorem: For rational coefficients, if a + √b is a zero (with √b irrational), then a - √b is also a zero.
- Descartes' Rule of Signs: The number of positive real zeros equals the number of sign changes in f(x) coefficients; negative real zeros relate to f(-x) coefficients.
Zeros and Rational Roots
- Zero: A solution of the function f(x) = 0.
- National Zero Theorem: Every rational zero of a polynomial with integer coefficients can be expressed as p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Polynomial Division and Factoring
- Factor Theorem: A polynomial f(x) has a factor x - k if and only if f(k) = 0.
- Remainder Theorem: The remainder of the division of f(x) by x - k is given by f(k).
- Synthetic Division: A method for dividing polynomials when the divisor is in the form (x - c).
- Long Division: Another method for dividing polynomials.
Special Factoring Techniques
- Factoring by Grouping: Group pairs of terms sharing a common monomial factor to simplify the polynomial.
- Factoring Cubes: Identify the cube roots of terms; use S.O.A.P. (Sum, Opposite, Add) for cube factoring.
- Difference of Two Cubes: a³ - b³ = (a - b)(a² + ab + b²).
- Sum of Two Cubes: a³ + b³ = (a + b)(a² - ab + b²).
Standard Forms and Types of Polynomials
- Common Polynomial Functions:
- Degree 2 (Quadratic): Standard form is f(x) = ax² + bx + c.
- Degree 1 (Linear): Standard form is f(x) = ax + b.
Types of Factorization
- Common Monomial Factor: Example: 8x² + 20x = 4x(2x + 5).
- Difference of Two Squares: Example: 9x² - 1 = (3x + 1)(3x - 1).
- Perfect Square Trinomial: Example: x² + 8x + 16 = (x + 4)².
- General Trinomial: Example: 2x² - 3x - 20 = (2x + 5)(x - 4).
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Description
Test your understanding of key concepts related to polynomial functions, including local maxima, minima, and turning points. Explore the Fundamental Theorem of Algebra and various zero theorems to solidify your knowledge about polynomial behavior and roots.
