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

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 $(ax + b)^2 = a^2x^2 + 2abx + b^2$.

2. Analyzing Each Polynomial

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

   • $a^2 = 49$ → $a = 7$
   • $b^2 = 16$ → $b = 4$
   • Check $2ab = 2(7)(4) = 56$ (this should match $-28$, which it does not).
   • Conclusion: Not a perfect square.

2. $9a^2 - 30a + 25$:

   • $a^2 = 9$ → $a = 3$
   • $b^2 = 25$ → $b = 5$
   • Check $2ab = 2(3)(5) = 30$ (this should match $-30$, which it does).
   • Conclusion: This is a perfect square, specifically $(3a - 5)^2$.

3. $25b^2 - 45b - 81$:

   • $a^2 = 25$ → $a = 5$
   • $b^2 = -81$ (not a perfect square).
   • Conclusion: Not a perfect square.

4. $16x^2 - 24x - 9$:

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

3. Final Conclusion

The only polynomial that is a perfect square trinomial is:

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