Polynomial Functions and Theorems
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Questions and Answers

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?

  • 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?

    The graph of every polynomial function of degree n has at most n-1 turning points.

    What does the Fundamental Theorem of Algebra state?

    <p>If f(x) is a polynomial of degree n where n&gt;0, then the equation f(x) = 0 has at least one solution in the set of complex numbers.</p> Signup and view all the answers

    What does the Fundamental Theorem of Algebra: Corollary specify?

    <p>If f(x) is a polynomial of degree n where n&gt;0, then the equation f(x)=0 has exactly n solutions, counting multiplicities.</p> Signup and view all the answers

    What is the Complex Conjugate Theorem?

    <p>If f is a polynomial function with real coefficients, and a + bi is an imaginary zero of f, then a - bi is also a zero of f.</p> Signup and view all the answers

    What does the Irrational Conjugates Theorem state?

    <p>If a + √b is a zero of f, then a - √b is also a zero of f.</p> Signup and view all the answers

    What does Descartes' Rule of Signs indicate?

    <p>The number of positive real zeros of f equals the number of changes in sign of the coefficients of f(x) or is less by an even number.</p> Signup and view all the answers

    What is the definition of a zero of a function?

    <p>A solution.</p> Signup and view all the answers

    What does The National Zero Theorem state?

    <p>If f(x)=anx^n +...+ a, with integer coefficients, then every rational zero of f takes the form p/q = factor of constant term a / factor of leading coefficient an.</p> Signup and view all the answers

    Give examples of rational numbers.

    <p>±5, 7/2, 1.75.</p> Signup and view all the answers

    What are Real Numbers?

    <p>Non-imaginary (no i).</p> Signup and view all the answers

    What does the Factor Theorem state?

    <p>A polynomial f(x) has a factor x - k if and only if f(k) = 0.</p> Signup and view all the answers

    What is the Remainder Theorem?

    <p>If a polynomial f(x) is divided by x - k, then the remainder is r = f(k).</p> Signup and view all the answers

    What is Synthetic Division?

    <p>A method of dividing polynomials, where the divisor is of the form (x - c).</p> Signup and view all the answers

    What is Long Division in the context of polynomials?

    <p>One method of dividing polynomials.</p> Signup and view all the answers

    What is Factoring by Grouping?

    <p>For some polynomials, you can factor by grouping pairs of terms that have a common monomial factor.</p> Signup and view all the answers

    What are the steps involved in factoring cubes?

    <ol> <li>Find the cube root of first term (=a), 2. Find the cube root of the second term (=b), 3. Plug into S.O.A.P.</li> </ol> Signup and view all the answers

    What are the steps in factoring a polynomial?

    <ol> <li>Standard form? 2. Reduce? 3. Factor first term. 4. Check your signs. 5. Factor last term. 6. Check!</li> </ol> Signup and view all the answers

    What is the Difference of Two Cubes factoring pattern?

    <p>a^3 - b^3 = (a - b)(a^2 + ab + b^2).</p> Signup and view all the answers

    What is the Sum of Two Cubes factoring pattern?

    <p>a^3 + b^3 = (a + b)(a^2 - ab + b^2).</p> Signup and view all the answers

    What does it mean for a polynomial to be factored completely?

    <p>A factorable polynomial with integer coefficients is factored completely if it is written as a product of unfactorable polynomials with integer coefficients.</p> Signup and view all the answers

    What is a common monomial factor?

    <p>8x^2 + 20x = 4x(2x + 5).</p> Signup and view all the answers

    What is the Difference of Two Squares pattern?

    <p>9x^2 - 1 = (3x + 1)(3x - 1).</p> Signup and view all the answers

    What is a Perfect Square Trinomial?

    <p>x^2 + 8x + 16 = (x + 4)^2.</p> Signup and view all the answers

    What is a General Trinomial?

    <p>2x^2 - 3x - 20 = (2x + 5)(x - 4).</p> Signup and view all the answers

    What is a Common Polynomial Function of Degree 2?

    <p>Type: Quadratic, Standard Form: f(x) = ax^2 + bx + c.</p> Signup and view all the answers

    What is a Common Polynomial Function of Degree 1?

    <p>Type: Linear.</p> Signup and view all the answers

    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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    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.

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