Polynomial functions are algebraic expressions with variables raised to non-negative integer exponents. This overview covers graphing techniques, identifying roots, turning points, y-intercepts, and end behavior. It also discusses constructing polynomial graphs and deducing equations from graphical representations, highlighting the importance of understanding polynomial functions in algebra and calculus.
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Polynomial functions are composed of terms with variables and coefficients
Descending order of variable powers
Polynomial functions are conventionally expressed in descending order of variable powers
Degree as the highest power of the variable
The degree of a polynomial function indicates the highest power of the variable
Polynomial functions do not include negative exponents or variables in the denominator
Graphing polynomial functions requires identifying roots, turning points, y-intercepts, and end behavior
Methods for finding roots
Roots can be found by setting the polynomial to zero and solving for x using factoring, the quadratic formula, or synthetic division
Multiplicity of roots
A root with multiplicity greater than one indicates a touch or bounce on the x-axis
Turning points can be found by calculating the derivative and solving for when it equals zero
The y-intercept can be found by evaluating the polynomial at x = 0, and the end behavior is influenced by the leading term
Polynomial graphs can be linear, quadratic, cubic, quartic, or quintic, depending on the degree
The degree of a polynomial determines the maximum number of direction changes and real roots, allowing for a rough sketch of the graph
Identifying x-intercepts and multiplicity
The x-intercepts and their multiplicity correspond to the factors of the polynomial
Inferring the leading coefficient
The leading coefficient can be inferred from the y-intercept and end behavior
The process of deducing the equation from the graph is a valuable skill for understanding the relationship between polynomial functions and their graphical representations
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The highest power of the variable in a polynomial, which is a non-negative integer, is known as the ______ of the polynomial.
degree
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Roots of a Polynomial
Values of x where the polynomial equals zero; found by factoring, quadratic formula, or synthetic division.
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Multiplicity of a Root
Indicates graph's behavior at x-axis: single root crosses, multiple root touches or flattens.
