Problem

Use the displayed diagrams to interpret \(A=(x+2)(x+7)\) as rectangle area for \(x>-2\). State the factor meanings and units, find both algebraic zeros, and explain why neither represents a valid positive-area rectangle.

Source rectangle model for A=(x+2)(x+7) square units: side lengths are x+2 units and x+7 units with domain x>−2; no zero-validity conclusion is shown.

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What this problem is really about

The two factors represent perpendicular side lengths, so their product represents area and carries square units. We’ll solve each factor for every algebraic zero, then apply the strict domain and positive-dimension requirements to decide whether any zero can describe a valid rectangle.

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Factors x + 2 and x + 7 are side lengths in units; their product is area. The zero x = −7 is selected as a valid rectangle dimension because it makes one factor zero.

Factors x + 2 and x + 7 are side lengths in units; their product is area. The zero x = −2 is accepted as a valid positive-area rectangle because one side is zero.

Factors x + 2 and x + 7 have unit lengths; their operation adds the side lengths, so A is a length. Zeros −2 and −7 are valid length values.

Factors x + 2 and x + 7 are side lengths in units; their product is area. With x > −2, algebraic zeros −2 and −7 do not give a valid positive-area rectangle.

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Use the displayed diagrams to interpret \(A=(x+2)(x+7)\) as rectangle area for \(x>-2\). State the factor meanings and units, find both algebraic zeros, and explain why neither represents a valid positive-area rectangle.

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