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Divide the following polynomials by the given monomials.

Let's go through the division of each polynomial by the given monomial one by one.

### (a) \((5x^2 - 6x)\) by \(3x\)

\[
\frac{5x^2 - 6x}{3x} = \frac{5x^2}{3x} - \frac{6x}{3x} = \frac{5}{3}x - 2
\]

### (b) \((3y^8 + 4y^6 + 5y^4)\) by \(y^4\)

\[
\frac{3y^8 + 4y^6 + 5y^4}{y^4} = \frac{3y^8}{y^4} + \frac{4y^6}{y^4} + \frac{5y^4}{y^4} = 3y^4 + 4y^2 + 5
\]

### (c) \(8(x^3y^2z^2 + x^2y^2z^2 + x^2y^2z^3)\) by \(4x^2y^2z^2\)

\[
\frac{8(x^3y^2z^2 + x^2y^2z^2 + x^2y^2z^3)}{4x^2y^2z^2} = 2\left(\frac{x^3y^2z^2}{x^2y^2z^2} + \frac{x^2y^2z^2}{x^2y^2z^2} + \frac{x^2y^2z^3}{x^2y^2z^2}\right)
\]
\[
= 2(x + 1 + z)
\]

### (d) \((x^3 + 2x^2 + 3x)\) by \(2x\)

\[
\frac{x^3 + 2x^2 + 3x}{2x} = \frac{x^3}{2x} + \frac{2x^2}{2x} + \frac{3x}{2x} = \frac{x^2}{2} + x + \frac{3}{2}
\]

### (e) \((p^3q^6 - p^6q^3)\) by \(p^3q^3\)

\[
\frac{p^3q^6 - p^6q^3}{p^3q^3} = \frac{p^3q^6}{p^3q^3} - \frac{p^6q^3}{p^3q^3} = q^3 - p^3
\]

### (f) \((32x^4y^4 - 16x^3y^4)\) by \((-8x^2y)\)

\[
\frac{32x^4y^4 - 16x^3y^4}{-8x^2y} = \frac{32x^4y^4}{-8x^2y} - \frac{16x^3y^4}{-8x^2y} = -4x^2y^3 + 2xy^3
\]

### (g) \(4(ax^2 + 2a^2x)\) by \(2ax\)

\[
\frac{4(ax^2 + 2a^2x)}{2ax} = 2\left(\frac{ax^2}{ax} + \frac{2a^2x}{ax}\right) = 2(x + 2a)
\]

### (h) \((38a^3b^3c^2 - 19a^4b^2c)\) by \(19a^2bc\)

\[
\frac{38a^3b^3c^2 - 19a^4b^2c}{19a^2bc} = 2\left(\frac{a^3b^3c^2}{a^2bc} - \frac{a^4b^2c}{a^2bc}\right)
\]
\[
= 2(a b^2 c - a^2 b)
\]

### (i) \((-14x^{12}y + 8x^5z)\) by \(2x^2\)

\[
\frac{-14x^{12}y + 8x^5z}{2x^2} = \frac{-14x^{12}y}{2x^2} + \frac{8x^5z}{2x^2} = -7x^{10}y + 4x^3z
\]

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Would you like more details on any specific step or explanation?

Here are 5 related questions to expand on this topic:
1. How do you handle division when there are negative exponents involved?
2. What is the general rule for dividing polynomials by monomials?
3. How would the process differ if you were dividing by a binomial instead?
4. How can factoring simplify polynomial division problems?
5. Why is it important to align terms based on degrees in polynomial division?

**Tip**: Always simplify each term individually when dividing a polynomial by a monomial to avoid confusion!

Learn how to divide polynomials by monomials with step-by-step solutions. This problem covers key algebraic concepts like polynomial division and the laws of exponents. Suitable for students in Grades 8-10.

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