To determine which polynomial is a perfect square trinomial, we will check each polynomial for the perfect square trinomial condition. A trinomial [tex]\(ax^2 + bx + c\)[/tex] is a perfect square trinomial if and only if [tex]\(b^2 = 4ac\)[/tex].
Let's evaluate each polynomial step by step:
1. Polynomial: [tex]\(25x^2 - 40x - 16\)[/tex]
- Here, [tex]\(a = 25\)[/tex], [tex]\(b = -40\)[/tex], and [tex]\(c = -16\)[/tex].
- Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (-40)^2 = 1600
\][/tex]
- Calculate [tex]\(4ac\)[/tex]:
[tex]\[
4ac = 4 \cdot 25 \cdot (-16) = 4 \cdot 25 \cdot (-16) = -1600
\][/tex]
- Compare [tex]\(b^2\)[/tex] with [tex]\(4ac\)[/tex]:
[tex]\[
1600 \neq -1600
\][/tex]
- This polynomial is not a perfect square trinomial.
2. Polynomial: [tex]\(9a^2 - 20a - 25\)[/tex]
- Here, [tex]\(a = 9\)[/tex], [tex]\(b = -20\)[/tex], and [tex]\(c = -25\)[/tex].
- Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (-20)^2 = 400
\][/tex]
- Calculate [tex]\(4ac\)[/tex]:
[tex]\[
4ac = 4 \cdot 9 \cdot (-25) = 4 \cdot 9 \cdot -25 = -900
\][/tex]
- Compare [tex]\(b^2\)[/tex] with [tex]\(4ac\)[/tex]:
[tex]\[
400 \neq -900
\][/tex]
- This polynomial is not a perfect square trinomial.
3. Polynomial: [tex]\(25b^2 - 15b + 9\)[/tex]
- Here, [tex]\(a = 25\)[/tex], [tex]\(b = -15\)[/tex], and [tex]\(c = 9\)[/tex].
- Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (-15)^2 = 225
\][/tex]
- Calculate [tex]\(4ac\)[/tex]:
[tex]\[
4ac = 4 \cdot 25 \cdot 9 = 4 \cdot 225 = 900
\][/tex]
- Compare [tex]\(b^2\)[/tex] with [tex]\(4ac\)[/tex]:
[tex]\[
225 \neq 900
\][/tex]
- This polynomial is not a perfect square trinomial.
4. Polynomial: [tex]\(16x^2 - 56x + 49\)[/tex]
- Here, [tex]\(a = 16\)[/tex], [tex]\(b = -56\)[/tex], and [tex]\(c = 49\)[/tex].
- Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (-56)^2 = 3136
\][/tex]
- Calculate [tex]\(4ac\)[/tex]:
[tex]\[
4ac = 4 \cdot 16 \cdot 49 = 3136
\][/tex]
- Compare [tex]\(b^2\)[/tex] with [tex]\(4ac\)[/tex]:
[tex]\[
3136 = 3136
\][/tex]
- This polynomial meets the condition for a perfect square trinomial.
Thus, the polynomial [tex]\(16x^2 - 56x + 49\)[/tex] is the perfect square trinomial among the given options.
