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Difficulty: MediumFactoring Polynomials

Which of the following is a factor of the polynomial 4x2−12xy+9y2−254x^2 - 12xy + 9y^2 - 25?

  1. 2x−3y−52x - 3y - 5Answer
  2. B
    2x+3y−52x + 3y - 5
  3. C
    2x−3y−252x - 3y - 25
  4. D
    4x−6y−54x - 6y - 5
  5. E
    2x−3y+252x - 3y + 25

Answer

The expression 2x−3y−52x - 3y - 5 is a factor of the polynomial.
The polynomial can be factored by first grouping the first three terms as a perfect square trinomial: 4x2−12xy+9y2=(2x−3y)24x^2 - 12xy + 9y^2 = (2x - 3y)^2. This simplifies the expression to (2x−3y)2−25(2x - 3y)^2 - 25. Since 25=5225 = 5^2, this is a difference of squares of the form A2−B2A^2 - B^2, which factors into (A−B)(A+B)(A - B)(A + B). Substituting A=2x−3yA = 2x - 3y and B=5B = 5 gives the factored form (2x−3y−5)(2x−3y+5)(2x - 3y - 5)(2x - 3y + 5). Thus, the option representing 2x−3y−52x - 3y - 5 is a factor.

Step-by-Step Solution

1
Group the first three terms of the polynomial and recognize the perfect square trinomial pattern.
4x2−12xy+9y2=(2x−3y)24x^2 - 12xy + 9y^2 = (2x - 3y)^2
The term 4x24x^2 is (2x)2(2x)^2, 9y29y^2 is (3y)2(3y)^2, and the middle term −12xy-12xy is −2(2x)(3y)-2(2x)(3y), which matches the perfect square trinomial identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
2
Rewrite the original expression using the factored trinomial and write 25 as a perfect square.
(2x−3y)2−52(2x - 3y)^2 - 5^2
Substituting the factored trinomial and expressing 25 as 525^2 sets up the expression as a difference of squares in the form A2−B2A^2 - B^2.
3
Apply the difference of squares factoring formula A2−B2=(A−B)(A+B)A^2 - B^2 = (A - B)(A + B).
(2x−3y−5)(2x−3y+5)(2x - 3y - 5)(2x - 3y + 5)
Substituting A=2x−3yA = 2x - 3y and B=5B = 5 into the formula yields the completely factored polynomial.

Key Concept

Factoring by grouping using perfect square trinomials and difference of squares
Estimated Time:1m 0s
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Topics

Factoring Polynomials

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