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Zorluk: ZorOperations on Polynomials

A square playground has a side length of 2x32x^3 meters. A square sandbox with a side length of x3−2xx^3 - 2x meters is built inside the playground. Which of the following expressions represents the area, in square meters, of the playground that is NOT covered by the sandbox?

  1. A
    3x6−4x4+4x23x^6 - 4x^4 + 4x^2
  2. B
    3x6−4x23x^6 - 4x^2
  3. C
    3x6+4x4+4x23x^6 + 4x^4 + 4x^2
  4. 3x6+4x4−4x23x^6 + 4x^4 - 4x^2Cevap
  5. E
    3x6+4x3−4x23x^6 + 4x^3 - 4x^2

Cevap

The correct expression is 3x6+4x4−4x23x^6 + 4x^4 - 4x^2.
The expression 3x6+4x4−4x23x^6 + 4x^4 - 4x^2 is correct because the area of the playground is (2x3)2=4x6(2x^3)^2 = 4x^6 and the area of the sandbox is (x3−2x)2=x6−4x4+4x2(x^3 - 2x)^2 = x^6 - 4x^4 + 4x^2. Subtracting the sandbox area from the playground area requires distributing the negative sign, resulting in 4x6−x6−(−4x4)−4x2=3x6+4x4−4x24x^6 - x^6 - (-4x^4) - 4x^2 = 3x^6 + 4x^4 - 4x^2.

Adım Adım Çözüm

1
Calculate the area of the square playground.
Areaplayground=(2x3)2=4x6\text{Area}_{\text{playground}} = (2x^3)^2 = 4x^6
The area of a square is equal to the square of its side length, and applying the exponent rules yields (2x3)2=22⋅(x3)2=4x6(2x^3)^2 = 2^2 \cdot (x^3)^2 = 4x^6.
2
Calculate the area of the square sandbox.
Areasandbox=(x3−2x)2=x6−4x4+4x2\text{Area}_{\text{sandbox}} = (x^3 - 2x)^2 = x^6 - 4x^4 + 4x^2
Using the binomial squaring formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, we square each term and compute the middle product, adding exponents for x3⋅x1=x4x^3 \cdot x^1 = x^4.
3
Subtract the sandbox's area from the playground's area.
4x6−(x6−4x4+4x2)=4x6−x6+4x4−4x2=3x6+4x4−4x24x^6 - (x^6 - 4x^4 + 4x^2) = 4x^6 - x^6 + 4x^4 - 4x^2 = 3x^6 + 4x^4 - 4x^2
Distribute the negative sign to all three terms inside the parentheses and combine the like terms of x6x^6.

Anahtar Kavram

Polynomial subtraction and squaring binomials with variables containing exponents.

Alternatif Yöntem

Evaluate the expression for a small integer value of xx. If x=2x = 2, the playground side length is 2(2)3=162(2)^3 = 16, giving an area of 162=25616^2 = 256. The sandbox side length is 23−2(2)=42^3 - 2(2) = 4, giving an area of 42=164^2 = 16. The remaining area is 256−16=240256 - 16 = 240. Substituting x=2x = 2 into the correct expression 3x6+4x4−4x23x^6 + 4x^4 - 4x^2 yields 3(64)+4(16)−4(4)=192+64−16=2403(64) + 4(16) - 4(4) = 192 + 64 - 16 = 240, confirming its correctness.
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Konular

Operations on Polynomials

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