Sure! Let's solve the given mathematical expression step-by-step.
We have the following expression to simplify:
[tex]\[ \frac{x^2 y-5 x^2}{y^3-3 y^2} \div \frac{y^2-2 y-8}{y^2-7 y+12}. \][/tex]
To simplify this, we need to perform the division of two fractions. Division of fractions can be converted to multiplication by taking the reciprocal of the divisor fraction.
So the given expression:
[tex]\[ \frac{x^2 y - 5 x^2}{y^3 - 3 y^2} \div \frac{y^2 - 2 y - 8}{y^2 - 7 y + 12} \][/tex]
can be rewritten as:
[tex]\[ \frac{x^2 y - 5 x^2}{y^3 - 3 y^2} \times \frac{y^2 - 7 y + 12}{y^2 - 2 y - 8}. \][/tex]
Now we multiply the numerators together and the denominators together:
[tex]\[ \frac{(x^2 y - 5 x^2) \times (y^2 - 7 y + 12)}{(y^3 - 3 y^2) \times (y^2 - 2 y - 8)}. \][/tex]
Next, let's factorize each polynomial where possible.
1. Factorize [tex]\( y^3 - 3 y^2 \)[/tex]:
[tex]\[ y^3 - 3 y^2 = y^2(y - 3). \][/tex]
2. Factorize [tex]\( y^2 - 7 y + 12 \)[/tex]:
[tex]\[ y^2 - 7 y + 12 = (y - 3)(y - 4). \][/tex]
3. Factorize [tex]\( y^2 - 2 y - 8 \)[/tex]:
[tex]\[ y^2 - 2 y - 8 = (y - 4)(y + 2). \][/tex]
So, the fraction becomes:
[tex]\[ \frac{(x^2 y - 5 x^2) \times \left( (y - 3)(y - 4) \right)}{y^2 (y - 3) \times (y - 4)(y + 2)}. \][/tex]
Now, we see that both the numerator and the denominator contain common factors that can be canceled out:
[tex]\[ \frac{x^2 (y - 5) \times \left( (y - 3)(y - 4) \right)}{y^2 (y - 3) \times (y - 4)(y + 2)}. \][/tex]
Cancelation of [tex]\((y - 3)\)[/tex] and [tex]\((y - 4)\)[/tex]:
[tex]\[ \frac{x^2 (y - 5)}{y^2 (y + 2)}.\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ \frac{x^2 (y - 5)}{y^2 (y + 2)}. \][/tex]
This is the final result.
