Divide the given polynomial by the given monomial: $(p^{3} q^{6} - p^{6} q^{3}) \div p^{3} q^{3}$

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Solution

Answer: (q³ – p³)
The four important terms used in the division operation are dividend, divisor, quotient and remainder. The formula to calculate the division of two numbers is:
Dividend ÷ Divisor = Quotient + Remainder.
Here,
The dividend is the number, which is being divided
The divisor is the number, which divides the number (dividend) into equal parts
The quotient is the result of the division operation
The remainder is the leftover number in the division operation.
$(p^{3} q^{6} - p^{6} q^{3}) \div p^{3} q^{3}$
$\Rightarrow \frac{(p^{3} q^{6} - p^{6} q^{3})}{p^{3} q^{3}} = \frac{p^{3} q^{3} (q^{3} - p^{3})}{p^{3} q^{3}} = (q^{3} - p^{3})$
Hence, $(q^{3} - p^{3})$ is the correct answer.

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If q is the mean proportional between p and r prove that :

$\frac{p^{3} + q^{3} + r^{3}}{p^{2} q^{2} r^{2}} = \frac{1}{p^{3}} + \frac{1}{q^{3}} + \frac{1}{r^{3}}$

(p^{3} q^{6} + p^{6} q^{3}) \div (p^{3} + q^{3})

p^{3} {(q - r)}^{3} + q^{3} {(r - p)}^{3} + r^{3} {(p - q)}^{3}

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