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Difficulty: MediumOperations on Polynomials

A rectangular garden has a length of 3x23x^2 meters and a width of 2x3−52x^3 - 5 meters. A square storage shed built inside the garden has a side length of x2−3x^2 - 3 meters. Which of the following polynomials represents the area of the remaining garden space, in square meters, that is not covered by the shed?

  1. A
    6x5−x4−15x2−96x^5 - x^4 - 15x^2 - 9
  2. B
    6x6−x4−9x2−96x^6 - x^4 - 9x^2 - 9
  3. C
    6x5−x4−21x2+96x^5 - x^4 - 21x^2 + 9
  4. 6x5−x4−9x2−96x^5 - x^4 - 9x^2 - 9Answer
  5. E
    6x5−x4−9x2+96x^5 - x^4 - 9x^2 + 9

Answer

The polynomial 6x5−x4−9x2−96x^5 - x^4 - 9x^2 - 9 represents the remaining area.
To find the remaining area, subtract the area of the square shed from the area of the rectangular garden. The area of the garden is 3x2(2x3−5)=6x5−15x23x^2(2x^3 - 5) = 6x^5 - 15x^2. The area of the shed is (x2−3)2=x4−6x2+9(x^2 - 3)^2 = x^4 - 6x^2 + 9. Subtracting the two gives (6x5−15x2)−(x4−6x2+9)=6x5−15x2−x4+6x2−9=6x5−x4−9x2−9(6x^5 - 15x^2) - (x^4 - 6x^2 + 9) = 6x^5 - 15x^2 - x^4 + 6x^2 - 9 = 6x^5 - x^4 - 9x^2 - 9, which represents the correct remaining garden space.

Step-by-Step Solution

1
Calculate the area of the rectangular garden.
Areagarden=3x2(2x3−5)=6x5−15x2\text{Area}_{\text{garden}} = 3x^2(2x^3 - 5) = 6x^5 - 15x^2
The area of a rectangle is the product of its length and width. Applying the distributive property and the exponent rule for multiplication (xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}), we get 3x2⋅2x3=6x53x^2 \cdot 2x^3 = 6x^5 and 3x2⋅(−5)=−15x23x^2 \cdot (-5) = -15x^2.
2
Calculate the area of the square storage shed.
Areashed=(x2−3)2=x4−6x2+9\text{Area}_{\text{shed}} = (x^2 - 3)^2 = x^4 - 6x^2 + 9
The area of a square is the square of its side length. We expand (x2−3)2(x^2 - 3)^2 using the binomial square pattern (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, which gives (x2)2−2(x2)(3)+(−3)2=x4−6x2+9(x^2)^2 - 2(x^2)(3) + (-3)^2 = x^4 - 6x^2 + 9.
3
Subtract the area of the shed from the area of the garden and simplify.
Remaining Area=6x5−x4−9x2−9\text{Remaining Area} = 6x^5 - x^4 - 9x^2 - 9
We subtract the shed's area from the garden's area: (6x5−15x2)−(x4−6x2+9)(6x^5 - 15x^2) - (x^4 - 6x^2 + 9). Distributing the negative sign gives 6x5−15x2−x4+6x2−96x^5 - 15x^2 - x^4 + 6x^2 - 9. Combining like terms and writing in descending order yields 6x5−x4−9x2−96x^5 - x^4 - 9x^2 - 9.

Key Concept

Operations on Polynomials
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Topics

Operations on Polynomials

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