Find the quotient of 20, x, cubed, y, cubed, plus, 20, x, squared, y, squared20x

Question

3
y
3
+20x
2
y
2
divided by minus, 5, x, y−5xy.

Answer 1

To find the quotient of \( 20x^3y^3 + 20x^2y^2 \) divided by \( -5xy \), we start by simplifying the expression.

1. Factor the numerator:

   The expression \( 20x^3y^3 + 20x^2y^2 \) can be factored: \[ 20x^3y^3 + 20x^2y^2 = 20x^2y^2(xy + 1) \] This is done by factoring out the common term \( 20x^2y^2 \).

2. Set up the division:

   Now, we need to divide this factored form by \( -5xy \): \[ \frac{20x^2y^2(xy + 1)}{-5xy} \]

3. Simplify the fraction:

   We can simplify the coefficients: \[ \frac{20}{-5} = -4 \] The variables also simplify:

   • The \( x^2 \) in the numerator and \( x \) in the denominator gives us \( x^{2-1} = x^{1} \) or just \( x \).
   • The \( y^2 \) in the numerator and \( y \) in the denominator gives us \( y^{2-1} = y^{1} \) or just \( y \).

   Thus, the expression becomes: \[ -4xy(xy + 1) \]

4. Final expression:

   The final simplified quotient is: \[ -4x(xy + 1)y = -4xy(xy + 1) \]

So, the quotient of \( 20x^3y^3 + 20x^2y^2 \) divided by \( -5xy \) is:

\[ \boxed{-4xy(xy + 1)} \]