Let's factor the given expression completely:
[tex]\[ 2x^3y^4 - 8x^2y^3 + 6x y^2 \][/tex]
First, let's identify and factor out the greatest common factor (GCF) of the entire expression.
1. Extract the GCF of each term:
Each term in the expression has a factor of [tex]\(2\)[/tex], [tex]\(x\)[/tex], and [tex]\(y^2\)[/tex]:
[tex]\[ 2x^3y^4 - 8x^2y^3 + 6xy^2 \][/tex]
Factoring out [tex]\(2xy^2\)[/tex] from each term, we get:
[tex]\[ 2xy^2 \left(\frac{2x^3y^4}{2xy^2} - \frac{8x^2y^3}{2xy^2} + \frac{6xy^2}{2xy^2} \right) \][/tex]
which simplifies to:
[tex]\[ 2xy^2 \left(x^2y^2 - 4xy + 3 \right) \][/tex]
2. Factor the quadratic expression inside the parentheses:
We now consider the quadratic polynomial [tex]\(x^2y^2 - 4xy + 3\)[/tex].
To factor this quadratic polynomial, notice that it can be factored in the form of:
[tex]\[ (x y - a)(x y - b) \][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are roots of the quadratic equation [tex]\(x y^2 - 4xy + 3 = 0\)[/tex].
Since the quadratic polynomial [tex]\(x^2y^2 - 4xy + 3\)[/tex] is similar to a quadratic in the form [tex]\(z^2 - 4z + 3\)[/tex] where [tex]\(z = xy\)[/tex], it can be factored as:
[tex]\[ (xy - 3)(xy - 1) \][/tex]
3. Combine all parts to form the fully factored expression:
Therefore, the fully factored form of the given expression is:
[tex]\[ 2xy^2(xy - 3)(xy - 1) \][/tex]
So, the expression [tex]\(2x^3y^4 - 8x^2y^3 + 6xy^2\)[/tex] factors completely as:
[tex]\[ 2xy^2(x y - 3)(x y - 1) \][/tex]
This is the final factored form.
