Problem

Using the rectangle diagram and \(A=(x+4)(x+9)\), label the two side lengths with units and describe their relationship. Require both lengths to be positive to find the physical domain, then interpret the algebraic zero boundaries.

Neutral rectangle area model labeled area A in square units; side dimensions, factors, zeros, and domain conclusions are not shown.

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What this problem is really about

The factors are rectangle dimensions, so each carries length units and both must be positive at the same time. We’ll compare the side expressions, intersect their positivity conditions to obtain the physical domain, and distinguish the product’s algebraic zero boundaries from values that represent a nondegenerate rectangle.

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The side lengths are x + 4 and x + 9, but only x + 9 needs to be positive. Both physical lengths must be positive.

The factors are square-unit areas, so their product is in fourth-power units and x may be any real number. Factors in an area product represent lengths here.

Negative or zero side lengths are valid rectangles, so both zero values are valid dimensions. A positive-area rectangle requires both lengths to be positive.

The side lengths are x + 4 and x + 9 units, so the area is in square units. Both lengths require x > −4; −4 and −9 are algebraic zero boundaries, with −4 the nearest degenerate boundary.

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Using the rectangle diagram and \(A=(x+4)(x+9)\), label the two side lengths with units and describe their relationship. Require both lengths to be positive to find the physical domain, then interpret the algebraic zero boundaries.

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