2xy² - 4xyz + xz² necesito llegar a
(2xy - 2xz + √2 xz) (y - z - √2/2 z)

To factor \( 2xy^2 - 4xyz + xz^2 \) into \( (2xy - 2xz + \sqrt{2}xz)(y - z - \frac{\sqrt{2}}{2}z) \), we will attempt to match terms from the expanded form of the factored expression to the original polynomial.

### Step 1: Expanding the Target Expression

We begin by expanding the given target factorization \( (2xy - 2xz + \sqrt{2} xz)(y - z - \frac{\sqrt{2}}{2} z) \):

\[
(2xy - 2xz + \sqrt{2}xz)(y - z - \frac{\sqrt{2}}{2}z)
\]

Using the distributive property (FOIL method):

\[
= (2xy)(y) + (2xy)(-z) + (2xy)\left(-\frac{\sqrt{2}}{2}z\right)
\]
\[
+ (-2xz)(y) + (-2xz)(-z) + (-2xz)\left(-\frac{\sqrt{2}}{2}z\right)
\]
\[
+ (\sqrt{2}xz)(y) + (\sqrt{2}xz)(-z) + (\sqrt{2}xz)\left(-\frac{\sqrt{2}}{2}z\right)
\]

Now, calculating each term:

\[
= 2xy^2 - 2xyz - \sqrt{2}xyz - 2xyz + 2xz^2 + \sqrt{2}xz^2 + \sqrt{2}xyz - \sqrt{2}xz^2 - xz^2
\]

### Step 2: Simplifying the Expanded Form

Now, let's simplify by combining like terms:

\[
= 2xy^2 - 4xyz + xz^2
\]

### Step 3: Comparing with the Original Expression

The expanded form \( 2xy^2 - 4xyz + xz^2 \) matches the original polynomial exactly.

Thus, the factorization is correct:

\[
2xy^2 - 4xyz + xz^2 = (2xy - 2xz + \sqrt{2} xz)(y - z - \frac{\sqrt{2}}{2} z)
\]

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Let me know if you need further details on the steps.

### Follow-up questions:
1. How do we identify the correct factorization for a given polynomial?
2. What role does the distributive property play in expanding and factoring expressions?
3. Can every polynomial be factored into simple binomials?
4. How does recognizing patterns in terms simplify the factoring process?
5. What other methods exist for factoring complex polynomials?

### Tip:
When factoring polynomials, it's helpful to check the result by expanding the factors back out to ensure accuracy.

2xy² - 4xyz + xz² necesito llegar a (2xy - 2xz + √2 xz) (y - z - √2/2 z)

Learn how to factor the polynomial 2xy² - 4xyz + xz² into (2xy - 2xz + √2 xz)(y - z - √2/2 z) by using algebraic techniques such as the distributive property. A step-by-step solution is provided, focusing on simplifying and verifying the expanded form.

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