Consider the polynomial function \( f(x) = (x + 1)(x - 2)(x - 3) \).
(a) Find all zeros.
(b) The zeros divide \(\mathbb{R}\) into several open intervals. Determine the sign of \( f(x) \) for \( x \) in each interval.
Answer (2)
a) f(x) = 0 <=> x = -1 or x = 2 or x = 3; b) x∈(-oo, -1) => f(x) < 0; x∈(-1, 2) => f(x) > 0; x∈(2, 3) => f(x) < 0; x∈(3, +oo) => f(x) > 0;
The zeros of the function f ( x ) = ( x + 1 ) ( x − 2 ) ( x − 3 ) are x = − 1 , 2 , 3 . The function is negative on the intervals ( − ∞ , − 1 ) and ( 2 , 3 ) , and positive on the intervals ( − 1 , 2 ) and ( 3 , ∞ ) . Thus, the sign of f ( x ) varies with each interval defined by the zeros.
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