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Difficulty: Very hardOperations on Polynomials
Which of the following expressions is equivalent to the expression below?
(3x3−2y2)2−2(x3−4y2)(4x3−5y2)(3x^3 - 2y^2)^2 - 2(x^3 - 4y^2)(4x^3 - 5y^2)
  1. A
    x9+30x3y2−36y4x^9 + 30x^3y^2 - 36y^4
  2. B
    x6−54x3y2+44y4x^6 - 54x^3y^2 + 44y^4
  3. x6+30x3y2−36y4x^6 + 30x^3y^2 - 36y^4Answer
  4. D
    x6+42x3y2−36y4x^6 + 42x^3y^2 - 36y^4
  5. E
    −7x6+72x3y2−76y4-7x^6 + 72x^3y^2 - 76y^4

Answer

The expression x6+30x3y2−36y4x^6 + 30x^3y^2 - 36y^4.
To find the correct expression, we first square the binomial to obtain 9x6−12x3y2+4y49x^6 - 12x^3y^2 + 4y^4. Next, we multiply the two binomials (x3−4y2)(4x3−5y2)(x^3 - 4y^2)(4x^3 - 5y^2) to get 4x6−21x3y2+20y44x^6 - 21x^3y^2 + 20y^4, and distribute the factor of 2 to obtain 8x6−42x3y2+40y48x^6 - 42x^3y^2 + 40y^4. Finally, we subtract this from the squared binomial by distributing the negative sign to all terms: 9x6−12x3y2+4y4−8x6+42x3y2−40y49x^6 - 12x^3y^2 + 4y^4 - 8x^6 + 42x^3y^2 - 40y^4. Combining like terms yields x6+30x3y2−36y4x^6 + 30x^3y^2 - 36y^4.

Step-by-Step Solution

1
Expand the squared binomial (3x3−2y2)2(3x^3 - 2y^2)^2.
9x6−12x3y2+4y49x^6 - 12x^3y^2 + 4y^4
Use the binomial squaring formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, where a=3x3a = 3x^3 and b=2y2b = 2y^2.
2
Multiply the two binomials (x3−4y2)(4x3−5y2)(x^3 - 4y^2)(4x^3 - 5y^2).
4x6−21x3y2+20y44x^6 - 21x^3y^2 + 20y^4
Apply the FOIL method to expand the product: 4x6−5x3y2−16x3y2+20y44x^6 - 5x^3y^2 - 16x^3y^2 + 20y^4, and combine the like middle terms.
3
Multiply the resulting trinomial from Step 2 by the constant factor of 2.
8x6−42x3y2+40y48x^6 - 42x^3y^2 + 40y^4
Distribute the factor of 2 to each term of the simplified trinomial expression.
4
Subtract the expression in Step 3 from the expression in Step 1.
x6+30x3y2−36y4x^6 + 30x^3y^2 - 36y^4
Distribute the negative sign to all terms of the subtracted polynomial: 9x6−12x3y2+4y4−8x6+42x3y2−40y49x^6 - 12x^3y^2 + 4y^4 - 8x^6 + 42x^3y^2 - 40y^4, and combine the remaining like terms.

Key Concept

Operations on Polynomials
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Topics

Operations on Polynomials

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