Factorize the Expression: 14x²y³ + 21xy⁴ + 70x⁵y²

Decompose the following expression into factors:

$14x^2y^3+21xy^4+70x^5y^2$

To decompose the expression $14x^2y^3 + 21xy^4 + 70x^5y^2$, we'll proceed with the following steps:

• Step 1: Identify the greatest common factor of the coefficients: $14$, $21$, and $70$.
• Step 2: Determine the GCF of the variables by identifying the smallest powers of $x$ and $y$ present in all terms.
• Step 3: Factor the GCF out of the entire expression.

Now, let's work through each step:

Step 1: The coefficients are $14$, $21$, and $70$. The GCF of these numbers is $7$.

Step 2: Consider the powers of $x$: $x^2$, $x$, and $x^5$ are present. The smallest power is $x^1$.
For $y$, we have $y^3$, $y^4$, and $y^2$. The smallest power is $y^2$.

Step 3: Now, factor out $7xy^2$:

$14x^2y^3 + 21xy^4 + 70x^5y^2 = 7xy^2(2xy + 3y^2 + 10x^4)$

Each step confirms this factorization aligns with the expression $7xy^2(2xy + 3y^2 + 10x^4)$.

Therefore, the solution to the factorization problem is $7xy^2(2xy + 3y^2 + 10x^4)$.

Since this expression matches one of the provided choices, it is evident that the correct answer is "All answers are correct" as $7xy^2(y(2x+3y)+10x^4)$, $7xy^2(x(2y+10x^3)+3y^2)$, and $7xy^2(2xy+3y^2+10x^4)$ all equate to the correct factorization approach when simplified.