Intro to Polynomials
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This document discusses polynomial functions and how to graph them. It defines a polynomial as a sum of terms with non-negative integer exponents. Polynomial graphs are smooth curves that may be lines, parabolas, or higher-order curves. To graph a polynomial, one determines the end behavior from the leading term, finds the x-intercepts by setting the polynomial equal to 0, and uses intercepts and test points to plot the graph over intervals. Multiplicity of roots affects whether the graph crosses or is tangent to the x-axis at those points.
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A polynomial function is a function where all exponents are non-negative integers. It is of the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where n is a non-negative integer and the coefficients ai are real numbers. The degree of a polynomial is the highest exponent present, and the leading coefficient is the coefficient of the term with the highest degree. The end behavior of a polynomial function can be determined by looking at the sign of the leading coefficient and whether the highest degree is even or odd. A polynomial function has at most n real zeros, where n is its degree, and at most n-1 turning points.Graphing polynomials
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Intro to Polynomials
- 1. Polynomial Functions and Models Module 12
- 2. Polynomial FunctionsA polynomial of degree n is a function of the formP(x) = anxn + an-1xn-1 + ... + a1x + a0Where an 0. The numbers a0, a1, a2, . . . , an are called the coefficients of the polynomial. The a0is the constant coefficientorconstant term. The number an, the coefficient of the highest power, is the leading coefficient, and the term anxn is the leading term.
- 3. Graphs of Polynomial Functions and Nonpolynomial Functions
- 4. Graphs of PolynomialsGraphs smooth curveDegree greater than 2 ex. f(x) = x3These graphs will not have the following:Break or holeCorner or cuspGraphs are linesDegree 0 or 1 ex. f(x) = 3 or f(x) = x 5Graphs are parabolasDegree 2 ex. f(x) = x2 + 4x + 8
- 5. Even- and Odd-Degree Functions
- 7. Finding Zeros of a PolynomialZero- another way of saying solutionZeros of PolynomialsSolutionsPlace where graph crosses the x-axis (x-intercepts)Zeros of the function Place where f(x) = 0
- 8. Using the Graphing Calculator to Determine ZerosGraph the following polynomial function and determine the zeros.Before graphing, determine the end behavior and the numberof relative maxima/minima.In factored form:P(x) = (x + 2)(x 1)(x 3)族
- 9. MultiplicityIf (x-c)k, k 1, is a factor of a polynomial function P(x) and:K is evenThe graph is tangent to the x-axis at (c, 0)K is oddThe graph crosses the x-axis at (c, 0)
- 10. Multiplicityy = (x + 2)族(x 1)続油Answer.油油油2 is a root of multiplicity 2, and 1 is a root of multiplicity 3. 油These are the 5 roots:2, 油2, 油1, 油1, 油1.
- 11. Multiplicityy = x続(x + 2)4(x 3)5Answer.油油油0 is a root of multiplicity 3,-2 is a root of multiplicity 4, and 3 is a root of multiplicity 5. 油
- 12. To Graph a PolynomialUse the leading term to determine the end behavior.Find all its real zeros (x-intercepts). Set y = 0.Use the x-intercepts to divide the graph into intervals and choose a test point in each interval to graph.Find the y-intercept. Set x = 0.Use any additional information (i.e. turning points or multiplicity) to graph the function.