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## POLYNOMIALS

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**Polynomials**A polynomial is a function of the form where the are real numbers and n is a nonnegative integer. The domain of a polynomial function is the set of real numbers**The Degree of Polynomial Functions**The Degree, of a polynomial function in one variable is the largest power of x Example Below is a polynomial of degree 2 See Page 183 for a summary of the properties of polynomials of degree less than or equal to two**Properties of Polynomial Functions**The graph of a polynomial function is a smooth and continuous curve A smooth curve is one that contains no Sharp corners or cusps A polynomial function is continuousif its graph has no breaks, gaps or holes**Power Functions**A power function if degree n, is a function of the form where a is a real number, and n > 0 is an integer Examples (degree 4), (degree 7) , (degree 1)**Graphs of even power functions**The polynomial function is even if n 2 is even. The functions graphed above are even. Note as n gets larger the graph becomes flatter near the origin, between (-1, 1), but increases when x > 1 and when x < -1. As |x| gets bigger and bigger, the graph increases rapidly.**Properties of an even function**The domain of an even function is the set of real numbers Even functions are symmetric with the y-axis The graph of an even functioncontains the points (0, 0) (1,1) (-1, 1)**Graphs of odd power functions**The polynomial function is odd if n 3 is odd. The functions graphed above are odd. Note as n gets larger the graph becomes flatter near the origin, -1 < x <1 but increases when x > 1 or decreases when x < -1 . As |x| gets bigger and bigger, the graph increases for values of x greater than 1 and decreased rapidly for values of x less than or equal to -1.**Properties of an odd function**The domain of an odd function is the set of real numbers Odd functions are symmetric with the origin The graph of an odd functioncontains the points (0, 0) (1,1) (-1,-1)**Zeros of a polynomial function**• A real number ris a real zero of the polynomial f (x) if f (r) =0 • If r is a zero of the polynomial, then r is an x – intercept. • If r is a zero of the polynomial f (x) then f (x) = (x – r) p (x), where p (x) is a polynomial**The intercepts of a polynomial**• If r is an x – intercept of a polynomial x, then • f( r ) = 0 • If r is an x – intercept then either • 1. The graph crosses the x axis at r or • 2. The graph touches the x axis at r