Answer :
We want to factor completely the expression
[tex]$$
x^6y^2 + 2x^4y^2 - x^4y.
$$[/tex]
Step 1. Identify the Common Factor
First, we look for a common factor in all three terms. Notice that every term has at least a factor of [tex]$x^4$[/tex] and a factor of [tex]$y$[/tex]. So, we factor out [tex]$x^4y$[/tex]:
[tex]$$
x^6y^2 + 2x^4y^2 - x^4y = x^4y\left(\frac{x^6y^2}{x^4y} + \frac{2x^4y^2}{x^4y} - \frac{x^4y}{x^4y}\right).
$$[/tex]
Step 2. Simplify Each Term Inside the Parentheses
Divide each term by the common factor [tex]$x^4y$[/tex]:
1. For the first term:
[tex]$$
\frac{x^6y^2}{x^4y} = x^{6-4}y^{2-1} = x^2y.
$$[/tex]
2. For the second term:
[tex]$$
\frac{2x^4y^2}{x^4y} = 2y^{2-1} = 2y.
$$[/tex]
3. For the third term:
[tex]$$
\frac{x^4y}{x^4y} = 1.
$$[/tex]
So the expression inside the parentheses becomes:
[tex]$$
x^2y + 2y - 1.
$$[/tex]
Step 3. Write the Final Answer
Now, putting it all together, the completely factored form is:
[tex]$$
x^4y\left(x^2y + 2y - 1\right).
$$[/tex]
Thus, the completely factored form of the given expression is
[tex]$$
\boxed{x^4y\left(x^2y + 2y - 1\right)}.
$$[/tex]
[tex]$$
x^6y^2 + 2x^4y^2 - x^4y.
$$[/tex]
Step 1. Identify the Common Factor
First, we look for a common factor in all three terms. Notice that every term has at least a factor of [tex]$x^4$[/tex] and a factor of [tex]$y$[/tex]. So, we factor out [tex]$x^4y$[/tex]:
[tex]$$
x^6y^2 + 2x^4y^2 - x^4y = x^4y\left(\frac{x^6y^2}{x^4y} + \frac{2x^4y^2}{x^4y} - \frac{x^4y}{x^4y}\right).
$$[/tex]
Step 2. Simplify Each Term Inside the Parentheses
Divide each term by the common factor [tex]$x^4y$[/tex]:
1. For the first term:
[tex]$$
\frac{x^6y^2}{x^4y} = x^{6-4}y^{2-1} = x^2y.
$$[/tex]
2. For the second term:
[tex]$$
\frac{2x^4y^2}{x^4y} = 2y^{2-1} = 2y.
$$[/tex]
3. For the third term:
[tex]$$
\frac{x^4y}{x^4y} = 1.
$$[/tex]
So the expression inside the parentheses becomes:
[tex]$$
x^2y + 2y - 1.
$$[/tex]
Step 3. Write the Final Answer
Now, putting it all together, the completely factored form is:
[tex]$$
x^4y\left(x^2y + 2y - 1\right).
$$[/tex]
Thus, the completely factored form of the given expression is
[tex]$$
\boxed{x^4y\left(x^2y + 2y - 1\right)}.
$$[/tex]
