To determine which of the given polynomials is a perfect square trinomial, we need to check if any of them can be written in the form [tex]\((p(x))^2 = (mx+n)^2\)[/tex]. A perfect square trinomial can be generally expressed as [tex]\(ax^2 + bx + c\)[/tex] where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] satisfy [tex]\(b^2 = 4ac\)[/tex].
Let's evaluate each polynomial step-by-step to identify the perfect square trinomial.
### 1. [tex]\(49x^2 - 8x + 16\)[/tex]
To check if this is a perfect square trinomial, we need:
[tex]\[ b^2 - 4ac = 0 \][/tex]
Here, [tex]\( a = 49 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 16 \)[/tex].
Calculating the discriminant:
[tex]\[ (-8)^2 - 4(49)(16) = 64 - 3136 = -3072 \][/tex]
Since the discriminant is not zero, [tex]\(49x^2 - 8x + 16\)[/tex] is NOT a perfect square trinomial.
### 2. [tex]\(4a^2 - 10a + 25\)[/tex]
Similarly, evaluate:
[tex]\[ b^2 - 4ac = 0 \][/tex]
Here, [tex]\( a = 4 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 25 \)[/tex].
Calculating the discriminant:
[tex]\[ (-10)^2 - 4(4)(25) = 100 - 400 = -300 \][/tex]
Since the discriminant is not zero, [tex]\(4a^2 - 10a + 25\)[/tex] is NOT a perfect square trinomial.
### 3. [tex]\(25b^2 - 5b + 10\)[/tex]
Evaluate:
[tex]\[ b^2 - 4ac = 0 \][/tex]
Here, [tex]\( a = 25 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = 10 \)[/tex].
Calculating the discriminant:
[tex]\[ (-5)^2 - 4(25)(10) = 25 - 1000 = -975 \][/tex]
Since the discriminant is not zero, [tex]\(25b^2 - 5b + 10\)[/tex] is NOT a perfect square trinomial.
### 4. [tex]\(16x^2 - 8x + 1\)[/tex]
Evaluate:
[tex]\[ b^2 - 4ac = 0 \][/tex]
Here, [tex]\( a = 16 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 1 \)[/tex].
Calculating the discriminant:
[tex]\[ (-8)^2 - 4(16)(1) = 64 - 64 = 0 \][/tex]
Since the discriminant is zero, [tex]\(16x^2 - 8x + 1\)[/tex] is a perfect square trinomial.
To confirm, let's see if it can be factored as [tex]\( (mx + n)^2 \)[/tex]:
[tex]\[ 16x^2 - 8x + 1 = (4x - 1)^2 \][/tex]
Expanding:
[tex]\[ (4x - 1)^2 = 16x^2 - 8x + 1 \][/tex]
This confirms that [tex]\(16x^2 - 8x + 1\)[/tex] is a perfect square trinomial.
### Conclusion
The polynomial that is a perfect square trinomial is:
[tex]\[ 16x^2 - 8x + 1 \][/tex]
