Problem

Create a sign chart for \(f(x)=(x-2)(x-5)\). Order the zeros, determine each factor’s sign on the three intervals, and write the positive, negative, and zero sets in interval or set notation.

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What this problem is really about

The factor zeros partition the number line into intervals where each factor keeps a constant sign. We’ll order the zeros, determine both factor signs on one point in each interval, multiply those signs, and report the strict positive and negative intervals separately from the zero set.

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The function is positive on every nonzero input and never negative. Between 2 and 5 the factors have opposite signs, so the product is negative.

The signs are +, −, +, but the positive and negative sets include the zeros. Zeros are neither positive nor negative.

The function is positive on (2, 5) and negative outside. The two outside intervals have factors with matching signs, so they are positive.

The zeros are 2 and 5. The sign is positive on (−∞, 2), negative on (2, 5), and positive on (5, ∞); the zero set is 2, 5.

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Create a sign chart for \(f(x)=(x-2)(x-5)\). Order the zeros, determine each factor’s sign on the three intervals, and write the positive, negative, and zero sets in interval or set notation.

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