You're given three polynomial division problems where you need to divide a polynomial (with multiple terms) by a monomial. The key idea is to
divide each term of the polynomial individually by the monomial, then simplify each resulting term.
Let’s solve each one step-by-step.
---
Problem 1:
\[
\frac{30x^4y - 45x^3y^2 + 15x^2y^3}{5x^2y}
\]
Step 1: Divide each term in the numerator by the denominator \(5x^2y\).
- First term: \(\frac{30x^4y}{5x^2y}\)
- Coefficients: \(30 ÷ 5 = 6\)
- Variables: \(x^4 ÷ x^2 = x^{4-2} = x^2\), and \(y ÷ y = y^0 = 1\)
- Result: \(6x^2\)
- Second term: \(\frac{-45x^3y^2}{5x^2y}\)
- Coefficients: \(-45 ÷ 5 = -9\)
- Variables: \(x^3 ÷ x^2 = x\), \(y^2 ÷ y = y\)
- Result: \(-9xy\)
- Third term: \(\frac{15x^2y^3}{5x^2y}\)
- Coefficients: \(15 ÷ 5 = 3\)
- Variables: \(x^2 ÷ x^2 = 1\), \(y^3 ÷ y = y^2\)
- Result: \(3y^2\)
Final Answer:
\[
\boxed{6x^2 - 9xy + 3y^2}
\]
---
Problem 2:
\[
\frac{12a^4b - 8a^3b^2 + 18a^2b^3}{-2a^2b}
\]
Step 1: Divide each term by \(-2a^2b\).
- First term: \(\frac{12a^4b}{-2a^2b}\)
- Coefficients: \(12 ÷ (-2) = -6\)
- Variables: \(a^4 ÷ a^2 = a^2\), \(b ÷ b = 1\)
- Result: \(-6a^2\)
- Second term: \(\frac{-8a^3b^2}{-2a^2b}\)
- Coefficients: \(-8 ÷ (-2) = 4\)
- Variables: \(a^3 ÷ a^2 = a\), \(b^2 ÷ b = b\)
- Result: \(4ab\)
- Third term: \(\frac{18a^2b^3}{-2a^2b}\)
- Coefficients: \(18 ÷ (-2) = -9\)
- Variables: \(a^2 ÷ a^2 = 1\), \(b^3 ÷ b = b^2\)
- Result: \(-9b^2\)
Final Answer:
\[
\boxed{-6a^2 + 4ab - 9b^2}
\]
---
Problem 3:
\[
\frac{96wx^3y^2z^2 - 144w^3xy^2z^2}{12wxy^2z}
\]
Step 1: Divide each term by \(12wxy^2z\).
- First term: \(\frac{96wx^3y^2z^2}{12wxy^2z}\)
- Coefficients: \(96 ÷ 12 = 8\)
- Variables:
- \(w ÷ w = 1\)
- \(x^3 ÷ x = x^2\)
- \(y^2 ÷ y^2 = 1\)
- \(z^2 ÷ z = z\)
- Result: \(8x^2z\)
- Second term: \(\frac{-144w^3xy^2z^2}{12wxy^2z}\)
- Coefficients: \(-144 ÷ 12 = -12\)
- Variables:
- \(w^3 ÷ w = w^2\)
- \(x ÷ x = 1\)
- \(y^2 ÷ y^2 = 1\)
- \(z^2 ÷ z = z\)
- Result: \(-12w^2z\)
Final Answer:
\[
\boxed{8x^2z - 12w^2z}
\]
---
##
✔ Summary of Answers:
1. \(\boxed{6x^2 - 9xy + 3y^2}\)
2. \(\boxed{-6a^2 + 4ab - 9b^2}\)
3. \(\boxed{8x^2z - 12w^2z}\)
---
## 💡 Key Concept:
When dividing a polynomial by a monomial,
distribute the division — that is, divide each term of the polynomial separately by the monomial. Then simplify each term using exponent rules:
\[
\frac{x^m}{x^n} = x^{m-n}, \quad \text{and} \quad \frac{c}{d} = \text{coefficient quotient}.
\]
This is essentially the reverse of the distributive property!
Parent Tip: Review the logic above to help your child master the concept of dividing polynomials by monomials worksheets.
