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Find the factor of this perfect square trinomials

To factor the given perfect square trinomial \(49m^2 - 56m + 16\), we need to identify it as a square of a binomial.

1. Write down the trinomial:
   \[
   49m^2 - 56m + 16
   \]

2. Recognize the structure of a perfect square trinomial \(a^2 - 2ab + b^2\), which factors to \((a - b)^2\).

3. Identify \(a^2\) and \(b^2\):
   \[
   a^2 = 49m^2 \quad \Rightarrow \quad a = 7m
   \]
   \[
   b^2 = 16 \quad \Rightarrow \quad b = 4
   \]

4. Confirm that the middle term fits the form \(-2ab\):
   \[
   -2ab = -2 \cdot 7m \cdot 4 = -56m
   \]

Since the middle term \(-56m\) matches, we can conclude that the trinomial is indeed a perfect square trinomial. Therefore, it factors to:
\[
(7m - 4)^2
\]

So, the factored form of \(49m^2 - 56m + 16\) is:
\[
(7m - 4)^2
\]

Would you like further details or have any questions?

Here are some related questions:
1. What are the properties of perfect square trinomials?
2. How do you identify a perfect square trinomial?
3. Can perfect square trinomials have negative coefficients?
4. How do you expand a binomial square to verify it matches a given trinomial?
5. What is the general formula for factoring perfect square trinomials?
6. How do you factor trinomials that are not perfect squares?
7. What other methods can be used to factor trinomials?
8. How do you verify if a factored form is correct?

**Tip:** Always check the middle term of a trinomial to confirm it fits the pattern of a perfect square trinomial before factoring.

Learn how to factor the perfect square trinomial 49m^2 - 56m + 16 step-by-step. This problem involves recognizing and factoring perfect square trinomials using the formula a^2 - 2ab + b^2 = (a - b)^2. Suitable for high school students studying algebra.

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