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Determine which polynomial is a perfect square trinomial. 49x2 − 28x + 16 9a2 − 30a + 25 25b2 − 45b − 81 16x2 − 24x − 9

Question

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Solution

To determine which polynomial is a perfect square trinomial, we will analyze each option.

1. Definition of Perfect Square Trinomial

A perfect square trinomial is a polynomial of the form (ax+b)2=a2x2+2abx+b2 (ax + b)^2 = a^2x^2 + 2abx + b^2 .

2. Analyzing Each Polynomial

  1. 49x228x+16 49x^2 - 28x + 16 :

    • a2=49 a^2 = 49 a=7 a = 7
    • b2=16 b^2 = 16 b=4 b = 4
    • Check 2ab=2(7)(4)=56 2ab = 2(7)(4) = 56 (this should match 28-28, which it does not).
    • Conclusion: Not a perfect square.
  2. 9a230a+25 9a^2 - 30a + 25 :

    • a2=9 a^2 = 9 a=3 a = 3
    • b2=25 b^2 = 25 b=5 b = 5
    • Check 2ab=2(3)(5)=30 2ab = 2(3)(5) = 30 (this should match 30-30, which it does).
    • Conclusion: This is a perfect square, specifically (3a5)2 (3a - 5)^2 .
  3. 25b245b81 25b^2 - 45b - 81 :

    • a2=25 a^2 = 25 a=5 a = 5
    • b2=81 b^2 = -81 (not a perfect square).
    • Conclusion: Not a perfect square.
  4. 16x224x9 16x^2 - 24x - 9 :

    • a2=16 a^2 = 16 a=4 a = 4
    • b2=9 b^2 = -9 (not a perfect square).
    • Conclusion: Not a perfect square.

3. Final Conclusion

The only polynomial that is a perfect square trinomial is:

  • 9a230a+25 9a^2 - 30a + 25 , which can be expressed as (3a5)2(3a - 5)^2.

This problem has been solved

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