Answer:
- use a graphing calculator, synthetic division, or direct evaluation
- limit of f(x) matches limit of x
Step-by-step explanation:
Often, an easy way to verify the zeros of a polynomial function is to graph it using technology. Modern graphing calculators make short work of this.
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Graphing
The given function is shown graphed in the attachment, with the zeros verified.
The given function is an odd-degree polynomial with a positive leading coefficient. Both the domain and the range of the function are "all real numbers." The positive leading coefficient means that f(x) will tend in the same direction as x when x gets large:
as x → -∞, f(x) → -∞
as x → ∞, f(x) → ∞
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Synthetic Division
If you want to evaluate the function by hand, there are several ways you can do this. One is to use synthetic division to both find the function value and reduce the polynomial to a lower degree for testing the next root.
The second screen shot shows the verification of the zero x=-7 using synthetic division. When the corresponding factor is divided from the polynomial, you get ...
f(x) = (x +7)(x² +x -6)
Your familiarity with factoring quadratics tells you the complete factorization is ...
f(x) = (x +7)(x +3)(x -2) . . . . . has zeros x = {-7, -3, 2}
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Evalutating the Function
Another way to verify the zeros is to evaluate the function at the proposed values of the zeros. Evaluating a polynomial is often made easier by writing it in Horner form.
f(x) = ((x +8)x +1)x -42
Then ...
f(-7) = ((-7 +8)(-7) +1)(-7) -42 = (-7 +1)(-7) -42 = 42 -42 = 0
f(2) = ((2 +8)(2) +1)(2) -42 = (20 +1)(2) -42 = 42 -42 = 0
f(-3) = ((-3 +8)(-3) +1)(-3) -42 = (-15 +1)(-3) -42 = 42 -42 = 0
