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Polynomial Functions

Polynomial Functions. A polynomial function is a function of the form f(x)= a n x n +a n-1 x n-1 +…a 1 x 1 +a 0. A power function. A power function of degree n is a function in the form f(x) = ax n where a is a real number, a ≠ 0 and n > 0 is an integer.

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Polynomial Functions

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  1. Polynomial Functions A polynomial function is a function of the form f(x)= anxn+an-1xn-1+…a1x1+a0

  2. A power function A power function of degree n is a function in the form f(x) = axnwhere a is a real number, a≠ 0 and n > 0 is an integer

  3. Properties of Power Functions when n is an even integer • The graph is symmetric with respect to the y axis • The domain is the set of all real numbers. The range is the set of nonnegative real numbers • The graph always contains the points (0,0), (1,1) and (-1,1) • As the exponent n increases in magnitude, the graph becomes more vertical when x<-1 or x>1; but for x near the origin, the graph tends to flatten out and lie closer to the x axis

  4. Properties of Power Functions when n is an odd integer • The graph is symmetric with respect to the origin • The domain and the range is the set of all real numbers • The graph always contains the points (0,0), (1,1), and (-1,-1) • As the exponent n increases in magnitude, the graph becomes more vertical when x<-1 or x>1; but for x near the origin the graph tends to flatten out and lie closer to the x axis

  5. If f is a polynomial function and r is a real number for which f(r) =0 then r is called a real zero of f, or root of f. If r is a real zero of f then r is an x intercept of the graph of f (x-r) is a factor of f. Example 1 Find a polynomial of degree 3 whose zeros are -3,2, and 5

  6. Multiplicity If the same factor x-r occurs more than once than r is called a repeated or multiple zero of f If (x-r)m is a factor of a polynomial f and (x-r)m+1 is not a factor of f, then r is called a zero multiplicity of m of f. Example 2 For the polynomial f(x) = 5(x-2)(x+3)2 (x-⅓)4 Identify the zeros and their multiplicity

  7. Graphing a Polynomial Using Its x intercepts Find the x and y intercepts of the graph Use the x intercepts to find the intervals on which the graph of f is above the x axis and the intervals on which the graph is below the x axis Locate other points on the graph and connect all the points plotted with a smooth continuous curve

  8. If r is a zero of Even Multiplicity Sign of f(x) does not change from one side to the other side of r Graph touches x axis of r If r is a zero of odd multiplicity Sign of f(x) changes from one side to the other side of r Graph crosses x axis at r. The points at which a graph changes direction are called turning points. In calculus such points are call local maxima and local minima Theorem: (Rule) If f is a polynomial function of degree n the f has at most (n-1) turning points

  9. End Behavior Theorem: (rule) For large values of x, either positive or negative, the graph of the polynomial f(x)= anxn+an-1xn-1+…a1x1+a0 resembles the graph of the power function y= axn

  10. The end behavior of a polynomial can only be of four types n≥2 even an >0 n≥2 even an< 0 n ≥3 odd an >0 n ≥3 odd an< 0

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