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Difficulty: MediumOperations on Polynomials

A homeowner has a square patio with a side length of 3x−23x - 2 feet. She decides to cover a portion of the patio with a rectangular outdoor rug that has dimensions 2x−12x - 1 feet by x+4x + 4 feet. Which of the following expressions represents the area, in square feet, of the patio that remains uncovered by the rug?

  1. A
    7x2−19x7x^2 - 19x
  2. B
    7x2−7x+87x^2 - 7x + 8
  3. 7x2−19x+87x^2 - 19x + 8Answer
  4. D
    9x2−21x+89x^2 - 21x + 8
  5. E
    7x2−5x7x^2 - 5x

Answer

7x2−19x+87x^2 - 19x + 8
The expression representing the uncovered area is obtained by subtracting the area of the rug from the area of the patio. First, the area of the square patio is calculated as (3x−2)2=9x2−12x+4(3x - 2)^2 = 9x^2 - 12x + 4. Second, the area of the rug is calculated as (2x−1)(x+4)=2x2+7x−4(2x - 1)(x + 4) = 2x^2 + 7x - 4. Subtracting the rug's area from the patio's area requires distributing the negative sign: (9x2−12x+4)−(2x2+7x−4)=9x2−12x+4−2x2−7x+4(9x^2 - 12x + 4) - (2x^2 + 7x - 4) = 9x^2 - 12x + 4 - 2x^2 - 7x + 4. Combining like terms yields 7x2−19x+87x^2 - 19x + 8.

Step-by-Step Solution

1
Calculate the area of the square patio by squaring its side length.
Apatio=(3x−2)2=9x2−12x+4A_{\text{patio}} = (3x - 2)^2 = 9x^2 - 12x + 4
The area of a square is equal to the square of its side length: Area=s2\text{Area} = s^2.
2
Calculate the area of the rectangular rug by multiplying its length and width.
Arug=(2x−1)(x+4)=2x2+8x−x−4=2x2+7x−4A_{\text{rug}} = (2x - 1)(x + 4) = 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4
The area of a rectangle is equal to the product of its length and width: Area=l×w\text{Area} = l \times w.
3
Subtract the area of the rug from the area of the patio, distributing the negative sign to all terms of the rug's area.
Auncovered=(9x2−12x+4)−(2x2+7x−4)=9x2−12x+4−2x2−7x+4=7x2−19x+8A_{\text{uncovered}} = (9x^2 - 12x + 4) - (2x^2 + 7x - 4) = 9x^2 - 12x + 4 - 2x^2 - 7x + 4 = 7x^2 - 19x + 8
Subtracting a polynomial requires distributing the negative sign to every term inside the parentheses and then combining like terms.

Key Concept

Subtracting one polynomial from another requires distributing the negative sign to every term of the subtracted polynomial before combining like terms.
Estimated Time:1m 30s
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Topics

Operations on Polynomials

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