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Find the factor of this perfect square trinomials

To factor the given perfect square trinomial \( 25y^2 + 70yz + 49z^2 \), we can use the fact that a perfect square trinomial of the form \( a^2 + 2ab + b^2 \) can be factored as \( (a + b)^2 \).

Let's identify the components \( a \) and \( b \) in the given trinomial \( 25y^2 + 70yz + 49z^2 \):

1. \( 25y^2 \) is a perfect square, and \( \sqrt{25y^2} = 5y \).
2. \( 49z^2 \) is a perfect square, and \( \sqrt{49z^2} = 7z \).
3. The middle term \( 70yz \) can be written as \( 2 \times 5y \times 7z \).

Since \( 25y^2 + 70yz + 49z^2 \) fits the pattern \( a^2 + 2ab + b^2 \) with \( a = 5y \) and \( b = 7z \), we can factor it as:

\[ (5y + 7z)^2 \]

So, the factor of the given perfect square trinomial is:

\[ (5y + 7z)^2 \]

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you recognize a perfect square trinomial?
2. Can you factor \( 16x^2 + 24xy + 9y^2 \) in a similar way?
3. What happens if the middle term is negative, for example, \( 25y^2 - 70yz + 49z^2 \)?
4. How would you factor a trinomial that is not a perfect square?
5. Can you verify the factorization of \( 25y^2 + 70yz + 49z^2 \) by expanding \( (5y + 7z)^2 \)?
6. What are some real-world applications of factoring trinomials?
7. How do you solve equations involving perfect square trinomials?
8. How would you factor a perfect square trinomial with fractional coefficients?

**Tip:** When factoring trinomials, always check if it can be expressed as a perfect square trinomial first. This simplifies the process significantly.

Learn how to factor perfect square trinomials with a detailed example problem and step-by-step solution. This problem covers key algebraic concepts, including factoring perfect square trinomials using the formula (a + b)^2 = a^2 + 2ab + b^2. Suitable for high school students in Grades 9-12.

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