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The “zero” of a function is just the value at which a function touches the x-axis. Factored Polynomial It is easy to find the roots of a polynomial when it is in factored form! x 2 x 15 ( x 3)( x 5) 2 (x - 3) and (x + 5) are factors of the polynomial. (x - 3) and (x + 5) are factors of the polynomial. (x - 3)(x + 5) = 0 (we want to know where the polynomial crosses the x-axis) So (x – 3) = 0 and (x + 5) = 0 The zeros are x = 3, x = -5 Practice: Find the roots of the following factored polynomials. 1. y = (x-2)3(x+3)(x-4) 2. y = (x-5)(x+2)3(x-14)2 3. y = (x+3)(x-15)4 4. y = x2(x+6)(x-6) Sometimes the polynomial won’t be factored! Ex. y x 3 x 2 6 x 2nd → TRACE (CALC) → 2: zero Choose a point to the left of the zero. Then press ENTER. This arrow indicates that you’ve chosen a point to the left of the zero. Choose a point to the rightof the zero. Then press ENTER. This arrow indicates that you’ve chosen a point to the right of the zero. Press ENTER one more time! Find the zeros of the following polynomials: y x 10 x 28 x 6 x 45 4 3 2 y x 17 x 63 x 245 x 1372 4 3 2 Solutions y x 10 x 28 x 6 x 45 4 3 2 y 3 , 1, 5 y x 17 x 63 x 245 x 1372 4 3 2 y 7, 4 End Behavior The end behavior of a graph describes the far left and the far right portions of the graph. We can determine the end behaviors of a polynomial using the leading coefficient and the degree of a polynomial. First determine whether the degree of the polynomial is even or odd. 2 f ( x) 2x 3x 5 degree = 2 so it is even Next determine whether the leading coefficient is positive or negative. Leading coefficient = 2 so it is positive Leading Coefficient Degree Even Odd High→High Low→High Low→Low High→Low + − Find the end behavior of the following polynomials. a. f ( x ) 2 x 5 x 9 3 b. f ( x ) 4 x 2 x 6 x 3 4 2 c. f ( x ) 4 x 3 x 2 x 5 2 d. f ( x ) 3 x 2 x x 3 x 4 4 3 2