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Question

Difficulty: MediumMultiplying Polynomials

A rectangular garden has a length of (3x−2)(3x - 2) meters and a width of (2x+5)(2x + 5) meters. Inside the garden, a smaller rectangular paved patio is built with a length of (x+4)(x + 4) meters and a width of 2x2x meters. Which of the following expressions represents the total area of the garden that is NOT paved, in square meters?

  1. 4x2+3x−104x^2 + 3x - 10Answer
  2. B
    4x2+19x−104x^2 + 19x - 10
  3. C
    4x2−8x−104x^2 - 8x - 10
  4. D
    4x2−7x4x^2 - 7x

Answer

The expression representing the unpaved area is 4x2+3x−104x^2 + 3x - 10.
The correct answer is found by calculating the area of the entire garden and subtracting the area of the patio. First, multiply the dimensions of the garden using the distributive property (FOIL): (3x−2)(2x+5)=6x2+15x−4x−10=6x2+11x−10(3x - 2)(2x + 5) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10. Next, find the area of the patio by distributing 2x2x: (x+4)(2x)=2x2+8x(x + 4)(2x) = 2x^2 + 8x. Finally, subtract the patio's area from the total area: (6x2+11x−10)−(2x2+8x)(6x^2 + 11x - 10) - (2x^2 + 8x). Make sure to distribute the negative sign to both terms in the second polynomial: 6x2+11x−10−2x2−8x=4x2+3x−106x^2 + 11x - 10 - 2x^2 - 8x = 4x^2 + 3x - 10.

Step-by-Step Solution

1
Calculate the total area of the garden by multiplying its length and width.
(3x−2)(2x+5)=6x2+15x−4x−10=6x2+11x−10(3x - 2)(2x + 5) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10
The area of a rectangle is found by multiplying its length by its width.
2
Calculate the area of the paved patio by multiplying its length and width.
(x+4)(2x)=2x2+8x(x + 4)(2x) = 2x^2 + 8x
We need the area of the inner rectangle in order to subtract it from the total area.
3
Subtract the patio's area from the total garden area to find the unpaved area.
(6x2+11x−10)−(2x2+8x)=6x2+11x−10−2x2−8x=4x2+3x−10(6x^2 + 11x - 10) - (2x^2 + 8x) = 6x^2 + 11x - 10 - 2x^2 - 8x = 4x^2 + 3x - 10
The unpaved region is the difference between the total area and the paved area. Note that the negative sign must be distributed to both terms in the second polynomial.

Key Concept

Multiplying Polynomials
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Topics

Multiplying Polynomials

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