Problem

Build the complete sign chart for \(f(x)~=~(x~+~1)(x~-~3)\). State the zeros and multiplicities, the sign on every interval, and where the sign changes.

Prompt sign chart for (x+1)(x−3) with zero boundaries, open intervals, and representative test points.

Big Picture

What this problem is really about

Order every real zero on a number line, then use those boundaries to create open test intervals. One representative input determines the product sign throughout each interval, while the zeros themselves remain separate because the function equals zero there. Use multiplicity parity as a final check on whether the sign should change at each boundary.

Answer choices

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The zeros are included inside positive or negative intervals instead of listed as separate zero boundaries.

The sign stays positive after the simple zero at −1, so both outer intervals are positive.

The zeros are −1 and 3, both simple. f(x) is positive on (−∞, −1), negative on (−1, 3), and positive on (3, ∞); the sign changes at both zeros.

The zeros are −1 and 3, but the interval signs are negative, positive, negative.

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Build the complete sign chart for \(f(x)~=~(x~+~1)(x~-~3)\). State the zeros and multiplicities, the sign on every interval, and where the sign changes.

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