Question

4. Write the trinomial represented by each rectangle of algebra tiles. Then, determine the

dimensions of each rectangle.

Answer 1

Trinomial:
`\boldsymbol{\text{x}^2 + 3\text{x} - 4}`

Dimensions:
`\boldsymbol{\text{x}+4} \text{ by } \boldsymbol{\text{x}-1}`

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Step-by-step explanation:

The big green square is x units by x units. Its area is
`\text{x} * \text{x} = \text{x}^2` which represents x squared.

The long vertical green rectangles are x units tall and 1 unit across. The area of one such rectangle is x times 1 = x. Since we have four of these green rectangles, so far we have x+x+x+x = 4x in additional area.

So far the area is
`\text{x}^2+4\text{x}`

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All of the green areas mentioned so far are positive areas so to speak.

In contrast, the red areas are negative areas. They take away from the positive. You can think of it like a bank balance.

If a single green rectangle is x square units in area, then a red rectangle is negative x or -x

So we go from 4x to 4x-x = 3x

Then we subtract off 4 due to the 4 small squares

This gives a total area of
`\text{x}^2 + 3\text{x} - 4` and this represents the standard form of the polynomial area. It is a trinomial since it has 3 terms.

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Now onto the dimensions.

Check out the diagram below. The overall figure is x+4 units across the horizontal dimension, and x-1 units tall along the vertical dimension.

If you were to use the FOIL rule, then you would find that
`(\text{x}+4)(\text{x}-1) = \text{x}^2 + 3\text{x} - 4` which helps show how the two forms connect to one another.

This also helps show that area = length*width which is what we'd expect from any rectangular area.