IDNLearn.com is the place where your questions are met with thoughtful and precise answers. Our experts provide prompt and accurate answers to help you make informed decisions on any topic.

Select the polynomial that is a perfect square trinomial.

A. [tex]49 x^2-8 x+16[/tex]

B. [tex]4 a^2-10 a+25[/tex]

C. [tex]25 b^2-5 b+10[/tex]

D. [tex]16 x^2-8 x+1[/tex]


Sagot :

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]