Question

1. divide.

a) $\frac{15x^2 - 20x}{5x}$
b) $\frac{16m^2 + 20mn}{4m}$
c) $\frac{18k^2 - 9k}{9k}$
d) $\frac{12m + 18mn}{-6m}$
e) $\frac{1.4d^2 + 1.8dk - 1.6d}{2d}$

Explanation:

Part a)

Step1: Factor numerator

Factor out \(5x\) from \(15x^2 - 20x\): \(15x^2 - 20x = 5x(3x - 4)\)

Step2: Divide by \(5x\)

\(\frac{5x(3x - 4)}{5x}\) = \(3x - 4\) (Cancel \(5x\))

Part b)

Step1: Factor numerator

Factor out \(4m\) from \(16m^2 + 20mn\): \(16m^2 + 20mn = 4m(4m + 5n)\)

Step2: Divide by \(4m\)

\(\frac{4m(4m + 5n)}{4m}\) = \(4m + 5n\) (Cancel \(4m\))

Part c)

Step1: Factor numerator

Factor out \(9k\) from \(18k^2 - 9k\): \(18k^2 - 9k = 9k(2k - 1)\)

Step2: Divide by \(9k\)

\(\frac{9k(2k - 1)}{9k}\) = \(2k - 1\) (Cancel \(9k\))

Part d)

Step1: Factor numerator

Factor out \(-6m\) from \(12m + 18mn\): \(12m + 18mn = -6m(-2 - 3n)\) (Wait, better: \(12m + 18mn = 6m(2 + 3n)\), but denominator is \(-6m\). So factor numerator: \(12m + 18mn = 6m(2 + 3n)\))

Step2: Divide by \(-6m\)

\(\frac{6m(2 + 3n)}{-6m}\) = \(-(2 + 3n)\) = \(-2 - 3n\) (Cancel \(6m\) and consider sign)

Part e)

Step1: Factor numerator

Factor out \(0.2d\) from \(1.4d^2 + 1.8dk - 1.6d\): \(1.4d^2 + 1.8dk - 1.6d = 0.2d(7d + 9k - 8)\)

Step2: Divide by \(2d\)

Wait, denominator is \(2d\), numerator factored as \(0.2d(7d + 9k - 8)\). Wait, maybe factor numerator by \(0.2d\): \(1.4d^2 = 0.2d \times 7d\), \(1.8dk = 0.2d \times 9k\), \(-1.6d = 0.2d \times (-8)\). So numerator: \(0.2d(7d + 9k - 8)\). Denominator: \(2d = 0.2d \times 10\)? Wait, no: \(1.4d^2 + 1.8dk - 1.6d = 0.2d(7d + 9k - 8)\), denominator \(2d = 0.2d \times 10\)? Wait, better: divide each term by \(-6m\)? Wait, original denominator: \(-6m\). Wait, numerator: \(12m + 18mn\), denominator: \(-6m\). So split the fraction: \(\frac{12m}{-6m} + \frac{18mn}{-6m}\) = \(-2 - 3n\). Yes, that's simpler. So:

Step1: Split the fraction

\(\frac{12m}{-6m} + \frac{18mn}{-6m}\)

Step2: Simplify each term

\(\frac{12m}{-6m} = -2\), \(\frac{18mn}{-6m} = -3n\). So total: \(-2 - 3n\)

Part e)

Step1: Factor numerator

Factor out \(0.2d\) from \(1.4d^2 + 1.8dk - 1.6d\): \(1.4d^2 + 1.8dk - 1.6d = 0.2d(7d + 9k - 8)\)

Step2: Divide by \(2d\)

Wait, denominator is \(2d\). Let's split the fraction: \(\frac{1.4d^2}{2d} + \frac{1.8dk}{2d} - \frac{1.6d}{2d}\)

Step3: Simplify each term

\(\frac{1.4d^2}{2d} = 0.7d\), \(\frac{1.8dk}{2d} = 0.9k\), \(\frac{1.6d}{2d} = 0.8\). So total: \(0.7d + 0.9k - 0.8\) (or multiply numerator and denominator by 10 to eliminate decimals: \(\frac{14d^2 + 18dk - 16d}{20d}\) = \(\frac{2d(7d + 9k - 8)}{20d}\) = \(\frac{7d + 9k - 8}{10}\) = \(0.7d + 0.9k - 0.8\))

Answer:

s:
a) \(3x - 4\)
b) \(4m + 5n\)
c) \(2k - 1\)
d) \(-2 - 3n\)
e) \(0.7d + 0.9k - 0.8\) (or \(\frac{7d + 9k - 8}{10}\))