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Difficulty: MediumOperations on Polynomials

An artist is designing a rectangular stained-glass window. The total area of the window, in square inches, is represented by the polynomial 8x2+2x−38x^2 + 2x - 3. The area of the central blue glass section, in square inches, is represented by the product (2x−3)(x−2)(2x - 3)(x - 2). The remaining portion of the window is made of clear glass. Which of the following expressions represents the area, in square inches, of the clear glass section?

  1. 6x2+9x−96x^2 + 9x - 9Answer
  2. B
    6x2−5x+36x^2 - 5x + 3
  3. C
    6x2+9x+36x^2 + 9x + 3
  4. D
    8x2+7x−98x^2 + 7x - 9
  5. E
    6x2+2x−96x^2 + 2x - 9

Answer

The expression representing the area of the clear glass section is 6x2+9x−96x^2 + 9x - 9.
The correct expression is 6x2+9x−96x^2 + 9x - 9. The area of the clear glass is found by expanding (2x−3)(x−2)(2x - 3)(x - 2) to get 2x2−7x+62x^2 - 7x + 6 and then subtracting this from the total area: (8x2+2x−3)−(2x2−7x+6)=8x2+2x−3−2x2+7x−6=6x2+9x−9(8x^2 + 2x - 3) - (2x^2 - 7x + 6) = 8x^2 + 2x - 3 - 2x^2 + 7x - 6 = 6x^2 + 9x - 9.

Step-by-Step Solution

1
Expand the product representing the area of the blue glass section using binomial multiplication.
(2x−3)(x−2)=2x2−4x−3x+6=2x2−7x+6(2x - 3)(x - 2) = 2x^2 - 4x - 3x + 6 = 2x^2 - 7x + 6
Converting the factored area of the blue glass section to standard form is necessary before subtraction.
2
Set up the subtraction of the blue glass area from the total area.
(8x2+2x−3)−(2x2−7x+6)(8x^2 + 2x - 3) - (2x^2 - 7x + 6)
The clear glass area is found by subtracting the blue glass area from the total window area.
3
Distribute the negative sign to each term of the second polynomial and combine like terms.
8x2+2x−3−2x2+7x−6=(8x2−2x2)+(2x+7x)+(−3−6)=6x2+9x−98x^2 + 2x - 3 - 2x^2 + 7x - 6 = (8x^2 - 2x^2) + (2x + 7x) + (-3 - 6) = 6x^2 + 9x - 9
Distributing the subtraction sign changes the signs of all terms inside the second set of parentheses, allowing for correct simplification.

Key Concept

Operations on Polynomials
Estimated Time:1m 30s
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Topics

Operations on Polynomials

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