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The given expression is:

\[
16x^6 - 56x^4 + 49x^2
\]

We want to check whether this is a perfect square trinomial. A perfect square trinomial takes the form:

\[
(a^2 + 2ab + b^2) = (a + b)^2
\]
or
\[
(a^2 - 2ab + b^2) = (a - b)^2
\]

Let's try factoring the given expression.

1. **Factor out the common term**: 
   Notice that all terms share a factor of \( x^2 \). Let's factor that out:

   \[
   x^2(16x^4 - 56x^2 + 49)
   \]

2. **Factor the quadratic inside the parentheses**: 
   Now, focus on factoring \( 16x^4 - 56x^2 + 49 \). This is a quadratic in terms of \( x^2 \), so we can treat it like a standard quadratic trinomial. 

   The expression \( 16x^4 - 56x^2 + 49 \) can be factored as:

   \[
   (4x^2 - 7)^2
   \]

   Why? Expanding \( (4x^2 - 7)^2 \) gives:

   \[
   (4x^2 - 7)(4x^2 - 7) = 16x^4 - 2(4x^2)(7) + 49 = 16x^4 - 56x^2 + 49
   \]

Thus, the original expression becomes:

\[
x^2(4x^2 - 7)^2
\]

So, the expression \( 16x^6 - 56x^4 + 49x^2 \) is a perfect square trinomial, and it factors as:

\[
x^2(4x^2 - 7)^2
\]

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How can we verify if an expression is a perfect square trinomial?
2. Can every trinomial be factored into a perfect square?
3. How do we factor higher-degree polynomials efficiently?
4. What is the general form of a perfect square trinomial?
5. How does factoring help in solving quadratic equations?

**Tip**: Always look for common factors first when simplifying or factoring any expression; this step often simplifies the overall problem.

Is the expression 16x^6 - 56x^4 + 49x^2 a perfect square trinomial?

Learn how to factor the expression 16x^6 - 56x^4 + 49x^2 and verify that it's a perfect square trinomial. This step-by-step guide explains the mathematical concepts, formulas, and theorems involved. Suitable for high school students in Grades 10-12.

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