Answer :
To solve the problem of dividing the expression \((20 x^8 y^3 - 12 x^5 y^2) \div (-4 x^2 y)\), we will break it down term by term.
### Step-by-Step Solution:
1. Divide Each Term Separately:
Let's start by dividing each term in the numerator by the denominator.
#### Term 1: \(\frac{20 x^8 y^3}{-4 x^2 y}\)
- Coefficients:
[tex]\[ \frac{20}{-4} = -5 \][/tex]
- Variables:
[tex]\[ x^8 \div x^2 = x^{8-2} = x^6 \][/tex]
[tex]\[ y^3 \div y = y^{3-1} = y^2 \][/tex]
- Combine:
[tex]\[ \frac{20 x^8 y^3}{-4 x^2 y} = -5 x^6 y^2 \][/tex]
#### Term 2: \(\frac{-12 x^5 y^2}{-4 x^2 y}\)
- Coefficients:
[tex]\[ \frac{-12}{-4} = 3 \][/tex]
- Variables:
[tex]\[ x^5 \div x^2 = x^{5-2} = x^3 \][/tex]
[tex]\[ y^2 \div y = y^{2-1} = y \][/tex]
- Combine:
[tex]\[ \frac{-12 x^5 y^2}{-4 x^2 y} = 3 x^3 y \][/tex]
2. Combine the Results:
Now that we have the simplified forms of each term, we combine them:
[tex]\[ -5 x^6 y^2 + 3 x^3 y \][/tex]
### Final Expression:
Thus, the result of dividing \((20 x^8 y^3 - 12 x^5 y^2) \div (-4 x^2 y)\) is:
[tex]\[ -5 x^6 y^2 + 3 x^3 y \][/tex]
Among the options provided:
- \(-5 x^6 y^2 + 3 x^3 y\) matches our solution.
So, the correct answer is:
[tex]\[ \boxed{-5 x^6 y^2 + 3 x^3 y} \][/tex]
### Step-by-Step Solution:
1. Divide Each Term Separately:
Let's start by dividing each term in the numerator by the denominator.
#### Term 1: \(\frac{20 x^8 y^3}{-4 x^2 y}\)
- Coefficients:
[tex]\[ \frac{20}{-4} = -5 \][/tex]
- Variables:
[tex]\[ x^8 \div x^2 = x^{8-2} = x^6 \][/tex]
[tex]\[ y^3 \div y = y^{3-1} = y^2 \][/tex]
- Combine:
[tex]\[ \frac{20 x^8 y^3}{-4 x^2 y} = -5 x^6 y^2 \][/tex]
#### Term 2: \(\frac{-12 x^5 y^2}{-4 x^2 y}\)
- Coefficients:
[tex]\[ \frac{-12}{-4} = 3 \][/tex]
- Variables:
[tex]\[ x^5 \div x^2 = x^{5-2} = x^3 \][/tex]
[tex]\[ y^2 \div y = y^{2-1} = y \][/tex]
- Combine:
[tex]\[ \frac{-12 x^5 y^2}{-4 x^2 y} = 3 x^3 y \][/tex]
2. Combine the Results:
Now that we have the simplified forms of each term, we combine them:
[tex]\[ -5 x^6 y^2 + 3 x^3 y \][/tex]
### Final Expression:
Thus, the result of dividing \((20 x^8 y^3 - 12 x^5 y^2) \div (-4 x^2 y)\) is:
[tex]\[ -5 x^6 y^2 + 3 x^3 y \][/tex]
Among the options provided:
- \(-5 x^6 y^2 + 3 x^3 y\) matches our solution.
So, the correct answer is:
[tex]\[ \boxed{-5 x^6 y^2 + 3 x^3 y} \][/tex]
