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?
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For any polynomial with a positive lead coefficient, as x → ∞, f(x) → ∞.
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For any polynomial with a positive lead coefficient, as x → -∞, f(x) → -∞.
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For polynomials of odd degree with a positive lead coefficient, as x → -∞, f(x) → -∞.
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For polynomials of odd degree with a negative lead coefficient, as x → -∞, f(x) → ∞.
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For polynomials of even degree with a positive lead coefficient, as x → -∞, f(x) → -∞.
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For polynomials of even degree with a negative lead coefficient, as x → -∞, f(x) → ∞.
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Polynomials of odd degree have what type of end behavior?
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Polynomials of even degree have what type of end behavior?
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What is a synonym for turning points?
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What is the domain of all polynomial functions?
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What is an increasing interval?
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What is a decreasing interval?
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What is a positive interval?
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What is a negative interval?
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What is the multiplicity of a zero?
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What does a multiplicity of 2 indicate?
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What is the x-intercept?
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What is the y-intercept?
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What is the difference between roots/zeros and x-intercepts?
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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.
