Characteristics of Polynomials Flashcards
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Questions and Answers

What is the degree of a polynomial?

The greatest degree of any term in the polynomial

The maximum number of turning points that a polynomial might have is equal to what?

The degree of the polynomial minus 1

The maximum number of real roots that a polynomial might have is equal to what?

The degree of the polynomial

The total number of roots (real or imaginary, including multiplicity) that a polynomial might have is equal to what?

<p>The degree of the polynomial</p> Signup and view all the answers

For any polynomial with a positive lead coefficient, as x → ∞, f(x) → ∞.

<p>True</p> Signup and view all the answers

For any polynomial with a positive lead coefficient, as x → -∞, f(x) → -∞.

<p>False</p> Signup and view all the answers

For polynomials of odd degree with a positive lead coefficient, as x → -∞, f(x) → -∞.

<p>True</p> Signup and view all the answers

For polynomials of odd degree with a negative lead coefficient, as x → -∞, f(x) → ∞.

<p>True</p> Signup and view all the answers

For polynomials of even degree with a positive lead coefficient, as x → -∞, f(x) → -∞.

<p>False</p> Signup and view all the answers

For polynomials of even degree with a negative lead coefficient, as x → -∞, f(x) → ∞.

<p>True</p> Signup and view all the answers

Polynomials of odd degree have what type of end behavior?

<p>Opposite</p> Signup and view all the answers

Polynomials of even degree have what type of end behavior?

<p>Same</p> Signup and view all the answers

What is a synonym for turning points?

<p>Relative (or local) extrema</p> Signup and view all the answers

What is the domain of all polynomial functions?

<p>(-∞, ∞)</p> Signup and view all the answers

What is an increasing interval?

<p>A portion of a function where the y-values are increasing over the interval</p> Signup and view all the answers

What is a decreasing interval?

<p>A portion of a function where the y-values are decreasing over the interval</p> Signup and view all the answers

What is a positive interval?

<p>A portion of a function where the y-values are greater than 0 for all x-values</p> Signup and view all the answers

What is a negative interval?

<p>A portion of a function where the y-values are less than 0 for all x-values</p> Signup and view all the answers

What is the multiplicity of a zero?

<p>The number of times the related linear factor is repeated in the factored form of the polynomial</p> Signup and view all the answers

What does a multiplicity of 2 indicate?

<p>Does not change from + to - or - to + (bounces)</p> Signup and view all the answers

What is the x-intercept?

<p>The x-coordinate of a point where a graph crosses the x-axis</p> Signup and view all the answers

What is the y-intercept?

<p>The y-coordinate of a point where a graph crosses the y-axis</p> Signup and view all the answers

What is the difference between roots/zeros and x-intercepts?

<p>Roots/zeros may be real or complex while x-intercepts are the real values for which the equation = 0</p> Signup and view all the answers

Study Notes

Degree of a Polynomial

  • Degree indicates the highest exponent in a polynomial.
  • Determines the polynomial's overall behavior.

Turning Points

  • Maximum turning points equal the polynomial degree minus one.

Roots of a Polynomial

  • Maximum number of real roots is equivalent to the polynomial's degree.
  • Total roots, counting multiplicity and including imaginary roots, also equals the degree.

End Behavior

  • For polynomials with a positive leading coefficient, as x approaches infinity, f(x) approaches infinity.
  • For these polynomials, as x approaches negative infinity, f(x) approaches negative infinity.
  • Odd degree polynomials with a positive lead coefficient will result in f(x) approaching negative infinity as x approaches negative infinity.
  • Odd degree polynomials with a negative lead coefficient will have f(x) approaching infinity as x approaches negative infinity.
  • Even degree polynomials with a positive lead coefficient yield f(x) approaching infinity as x approaches negative infinity.
  • Even degree polynomials with a negative lead coefficient result in f(x) approaching negative infinity as x approaches negative infinity.

End Behavior Patterns

  • Odd degree polynomials display opposite end behaviors on either side.
  • Even degree polynomials demonstrate the same end behavior at both extremes.

Turning Points Synonym

  • Turning points are also referred to as relative or local extrema, indicating local maxima or minima.

Domain of Polynomials

  • The domain of all polynomial functions is (-∞, ∞), signifying the function is defined for all real numbers.

Function Intervals

  • Increasing intervals are segments where y-values rise across the interval.
  • Decreasing intervals describe sections where y-values decline over the interval.
  • Positive intervals indicate portions where y-values surpass zero for all x-values.
  • Negative intervals show parts where y-values fall below zero for all x-values.

Multiplicity of Zeros

  • Multiplicity refers to how many times a linear factor occurs in a polynomial's factored format.
  • A zero with multiplicity of 2 results in no sign change at the root (graph "bounces" off the axis).

Intercepts

  • The x-intercept is where a graph intersects the x-axis, represented by the x-coordinate of that point.
  • The y-intercept indicates the point where a graph crosses the y-axis, described by the y-coordinate of that point.

Roots, Zeros, and X-Intercepts

  • Roots or zeros of a polynomial may be real or complex numbers.
  • X-intercepts specifically refer to the real values where the polynomial equation equals zero.

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Test your understanding of polynomial characteristics with these flashcards. Learn about polynomial degrees, turning points, and real roots through concise definitions. Perfect for students looking to solidify their algebra knowledge.

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