Factorize the Expression: (16xy/z + 40x/yz - 56x²/yz²)

Decompose the following expression into factors:

$\frac{16xy}{z}+\frac{40x}{yz}-\frac{56x^2}{yz^2}$

To solve this problem, we'll proceed with the following steps:

• Step 1: Identify the greatest common factor in the expression.
• Step 2: Factor out this common factor from each term.
• Step 3: Simplify the resulting expression inside the parentheses.

Now, let's work through each step:

Step 1: The expression given is
$\frac{16xy}{z}+\frac{40x}{yz}-\frac{56x^2}{yz^2}$.
First, identify the greatest common factor (GCF) from the numerators and the highest common power in the denominators:

• The coefficients of the terms are $16$, $40$, and $56$, with a GCF of $8$.
• The common variable in the numerators is $x$.
• The shared parts of the denominators are $z$.

Thus, the GCF for the entire expression with terms considered is:

$\frac{8x}{z}$.

Step 2: Factor out $\frac{8x}{z}$ from the expression:

$\frac{8x}{z} \left( \frac{16xy}{8x}-\frac{40x}{8x}y+\frac{56x^{2}}{8x}y^{2} \right) \equiv \frac{8x}{z}(2y + \frac{5}{y} - \frac{7x}{yz})$.

Step 3: The expression in the parentheses should now be simplified:
After factoring, the expression inside the parentheses becomes $2y+\frac{5}{y}-\frac{7x}{yz}$.

Therefore, the factored form of the original expression is:

$\frac{8x}{z}(2y+\frac{5}{y}-\frac{7x}{yz})$.