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

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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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