To determine which polynomial is a perfect square trinomial, we need to factor each polynomial and check if the factors can be expressed as the square of a binomial. Here are the polynomials given:
1. [tex]\(25x^2 - 40x - 16\)[/tex]
2. [tex]\(9a^2 - 20a - 25\)[/tex]
3. [tex]\(25b^2 - 15b + 9\)[/tex]
4. [tex]\(16x^2 - 56x + 49\)[/tex]
### 1. Polynomial [tex]\(25x^2 - 40x - 16\)[/tex]
First, let's attempt to factor [tex]\(25x^2 - 40x - 16\)[/tex]:
[tex]\[
25x^2 - 40x - 16
\][/tex]
Checking the factorization, this polynomial does not simplify to the square of a binomial. Thus, it is not a perfect square trinomial.
### 2. Polynomial [tex]\(9a^2 - 20a - 25\)[/tex]
Next, we factor [tex]\(9a^2 - 20a - 25\)[/tex]:
[tex]\[
9a^2 - 20a - 25
\][/tex]
Checking the factorization, this polynomial does not simplify to the square of a binomial. Thus, it is not a perfect square trinomial.
### 3. Polynomial [tex]\(25b^2 - 15b + 9\)[/tex]
For [tex]\(25b^2 - 15b + 9\)[/tex], let's try to factor it:
[tex]\[
25b^2 - 15b + 9
\][/tex]
Checking the factorization, this polynomial does not simplify to the square of a binomial. Thus, it is not a perfect square trinomial.
### 4. Polynomial [tex]\(16x^2 - 56x + 49\)[/tex]
Lastly, we factor [tex]\(16x^2 - 56x + 49\)[/tex]:
[tex]\[
16x^2 - 56x + 49 = (4x - 7)^2
\][/tex]
The factorization simplifies to [tex]\((4x - 7)^2\)[/tex], which is clearly the square of the binomial [tex]\(4x - 7\)[/tex].
Thus, the polynomial [tex]\(16x^2 - 56x + 49\)[/tex] is indeed a perfect square trinomial.
### Conclusion
The polynomial [tex]\(16x^2 - 56x + 49\)[/tex] is the perfect square trinomial. The other polynomials do not simplify to the squares of binomials and hence are not perfect square trinomials.
