Factor completely [tex][tex]$2x^3y^4 - 8x^2y^3 + 6xy^2$[/tex][/tex].

A. [tex][tex]$2\left(x^3 y^4 - 4x^2 y^3 + 3xy^2\right)$[/tex][/tex]

B. [tex][tex]$2x\left(x^2 y^4 - 4xy^3 + 3y^2\right)$[/tex][/tex]

C. [tex][tex]$2xy^2\left(x^2 y^2 - 4xy + 3\right)$[/tex][/tex]

D. Prime



Answer :

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.