To factor the expression [tex]\(2x^3 y^4 - 8x^2 y^3 + 6x y^2\)[/tex] completely, let's go through the step-by-step factoring process:
1. Find the Greatest Common Factor (GCF):
The expression is [tex]\(2x^3 y^4 - 8x^2 y^3 + 6x y^2\)[/tex].
We first identify the GCF of the coefficients [tex]\(2, -8,\)[/tex] and [tex]\(6\)[/tex], which is [tex]\(2\)[/tex].
Next, we find the GCF of the variables in each term. The smallest power of [tex]\(x\)[/tex] common in all terms is [tex]\(x\)[/tex], and for [tex]\(y\)[/tex], it is [tex]\(y^2\)[/tex].
Therefore, the GCF of the entire expression is [tex]\(2 x y^2\)[/tex].
2. Factor out the GCF:
We factor out [tex]\(2x y^2\)[/tex] from each term:
[tex]\[
2 x y^2 (x^2 y^2 - 4x y + 3)
\][/tex]
3. Factor the quadratic polynomial:
Now, we need to factor the quadratic polynomial [tex]\(x^2 y^2 - 4x y + 3\)[/tex]. Observe that it is a quadratic in [tex]\(xy\)[/tex]:
Let [tex]\(u = xy\)[/tex]. Thus, the expression becomes:
[tex]\[
x^2 y^2 - 4x y + 3 = u^2 - 4u + 3
\][/tex]
4. Factor the quadratic [tex]\(u^2 - 4u + 3\)[/tex]:
To factor [tex]\(u^2 - 4u + 3\)[/tex], we look for two numbers that multiply to [tex]\(3\)[/tex] and add to [tex]\(-4\)[/tex]. These numbers are [tex]\(-1\)[/tex] and [tex]\(-3\)[/tex]. Thus:
[tex]\[
u^2 - 4u + 3 = (u - 1)(u - 3)
\][/tex]
Substituting back [tex]\(u = xy\)[/tex], we get:
[tex]\[
(xy - 1)(xy - 3)
\][/tex]
5. Combine all the factors:
Now, combine the GCF [tex]\(2x y^2\)[/tex] with the factored quadratic function:
[tex]\[
2 x y^2 (xy - 1)(xy - 3)
\][/tex]
6. Write the final factored form:
The completely factored form of the given expression is:
[tex]\[
2 x y^2 (x y - 1)(x y - 3)
\][/tex]
This is the desired result. So, the fully factored expression is:
[tex]\[
2x y^2 (xy - 1)(xy - 3)
\][/tex]
