Factor polynomial $$ x^{2}y-25y+x^{2}z-25z $$

$$ x^{2}y-25y+x^{2}z-25z = (y+z)(x+5)(x-5) $$

Step 1 :

To factor $ x^2y-25y+x^2z-25z $ we can use factoring by grouping.

Group $ \color{blue}{ x^2y }$ with $ \color{blue}{ -25y }$ and $ \color{red}{ x^2z }$ with $ \color{red}{ -25z }$ then factor each group.

$$ \begin{aligned} x^2y-25y+x^2z-25z &= ( \color{blue}{ x^2y-25y } ) + ( \color{red}{ x^2z-25z }) = \\ &= \color{blue}{ y( x^2-25 )} + \color{red}{ z( x^2-25 ) } = \\ &= (y+z)(x^2-25) \end{aligned} $$

Step 2 :

Rewrite $ x^2-25 $ as:

$$ \color{blue}{ x^2-25 = (x)^2 - (5)^2 } $$

Now we can apply the difference of squares formula.

$$ I^2 - II^2 = (I - II)(I + II) $$

After putting $ I = x $ and $ II = 5 $ , we have:

$$ x^2-25 = (x)^2 - (5)^2 = ( x-5 ) ( x+5 ) $$

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