SECTION 4.1 Polynomial Functions 195 If a polynomial function f is factored completely, it is easy to locate the x -intercepts of the graph by solving the equation f x 0 ( ) = using the Zero-Product Property. For example, if f x x x 1 3 , 2 ( ) ( ) ( ) = − + then the solutions of the equation f x x x 1 3 0 2 ( ) ( ) ( ) = − + = are 1 and 3. − That is, f 1 0 ( ) = and f 3 0. ( ) − = Figure 6 Graph of a polynomial function x y Above x-axis Crosses x-axis Touches x-axis Crosses x-axis Below x-axis Below x-axis Below x-axis Above x-axis DEFINITION Real Zero If f is a function and r is a real number for which f r 0, ( ) = then r is called a real zero of f. • r is a real zero of a polynomial function f. • r is an x -intercept of the graph of f. • x r − is a factor of f. • r is a real solution to the equation f x 0. ( ) = As a consequence of this definition, the following statements are equivalent. So the real zeros of a polynomial function are the x -intercepts of its graph, and they are found by solving the equation f x 0. ( ) = Figure 7 f x x x x 3 2 5 ( ) ( )( )( ) = + − − Finding a Polynomial Function from Its Real Zeros (a) Find a polynomial function of degree 3 whose real zeros are 3, − 2, and 5. (b) Use a graphing utility to graph the polynomial found in part (a) to verify your result. EXAMPLE 4 (a) If r is a real zero of a polynomial function f, then x r − is a factor of f. This means that x x x 3 3, 2, ( ) − − = + − and x 5 − are factors of f. As a result, any polynomial function of the form f x a x x x 3 2 5 ( ) ( )( )( ) = + − − where a is a nonzero real number, qualifies. The value of a causes a stretch, compression, or reflection, but it does not affect the x -intercepts of the graph. Do you know why? (b) We choose to graph f with a 1. = Then f x x x x 3 2 5 ( ) ( )( )( ) = + − − Figure 7 shows the graph of f using Geogebra. Notice that the x -intercepts are 3, − 2, and 5. Solution
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