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Questions and Answers
What is the formula used to calculate the volume of a cereal box?
What is the formula used to calculate the volume of a cereal box?
V = lwh
What is the formula used to calculate the volume of a soda can?
What is the formula used to calculate the volume of a soda can?
V = πr^2h
What is a polynomial?
What is a polynomial?
A polynomial is an algebraic expression consisting of variables that can be added, subtracted, multiplied, and divided.
Name two operations with variables that cannot exist in a polynomial.
Name two operations with variables that cannot exist in a polynomial.
How are polynomials categorized by the number of terms?
How are polynomials categorized by the number of terms?
What determines the degree of a polynomial with one variable?
What determines the degree of a polynomial with one variable?
What is the standard way to write a polynomial?
What is the standard way to write a polynomial?
What is the degree of the polynomial 3x - 4?
What is the degree of the polynomial 3x - 4?
What is the degree of the polynomial -x^2 + 2x - 1?
What is the degree of the polynomial -x^2 + 2x - 1?
What does the degree of a polynomial indicate about its graph?
What does the degree of a polynomial indicate about its graph?
What should you pay close attention to when rearranging terms in a polynomial?
What should you pay close attention to when rearranging terms in a polynomial?
Classify the polynomial -6x^5 + 4x^3 + 3x^2 + 11.
Classify the polynomial -6x^5 + 4x^3 + 3x^2 + 11.
What is the result of writing the expression -2x^2(-5x^2 + 4x^3)?
What is the result of writing the expression -2x^2(-5x^2 + 4x^3)?
Consider the leading term of the polynomial function 2x^7 - 8x^6 - 3x^5 - 3. What is the end behavior?
Consider the leading term of the polynomial function 2x^7 - 8x^6 - 3x^5 - 3. What is the end behavior?
How many turning points are there for the graph of y = 2x - x^3?
How many turning points are there for the graph of y = 2x - x^3?
What is the degree of the polynomial that generates the given data? x: -6 / -3 / 0 / 3 / 6; y: 846 / 108 / 0 / -126 / -918.
What is the degree of the polynomial that generates the given data? x: -6 / -3 / 0 / 3 / 6; y: 846 / 108 / 0 / -126 / -918.
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Study Notes
Polynomial Functions Overview
- Volume formulas such as V = lwh for boxes and V = πr²h for cans are polynomial equations.
Definition and Operations
- A polynomial is an algebraic expression made of variables combined through addition, subtraction, multiplication, and division.
- Polynomials must not have variables in denominators or under radical symbols.
Classification of Polynomials
- Polynomials are classified by the number of terms:
- Monomial: one term
- Binomial: two terms
- Trinomial: three terms
- Classification also occurs by degree, determined by the highest exponent:
- Linear polynomials have degree 1.
- Quadratic polynomials have degree 2.
Standard Form
- Polynomials should be written in descending order of exponents.
- The degree of a polynomial indicates the influence of the term with the largest power on the graph.
Example Exponents and Degrees
- The polynomial 3x - 4 has a degree of 1 (greatest exponent is 1).
- The polynomial -x² + 2x - 1 has a degree of 2 (greatest exponent is 2).
Graph Characteristics
- The degree of a polynomial affects graph shape, including the number of turning points and end behavior.
- Turning points are where the graph changes direction; e.g., the vertex of a parabola.
- End behavior indicates the direction the graph extends towards infinity.
Rearranging Terms
- When creating a standard polynomial form, maintain the sign of each term; negatives must stay with their respective terms.
Polynomial Classifications and Calculations
- The polynomial -6x^5 + 4x^3 + 3x^2 + 11 is classified as quintic.
- Expanding -2x²(-5x² + 4x³) results in -8x^5 - 10x^4.
End Behavior in Graphs
- The leading term of 2x^7 in the polynomial indicates down and up end behavior, reflecting that n is odd and a is positive.
Turning Points in Graphs
- The function y = 2x - x³ exhibits two turning points, with the first point decreasing and the second point increasing.
Degree from Data
- Given the data points, the degree of the polynomial that fits is cubic.
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