Prove that: $(p + q)^{4} - (p - q)^{4} = 8 p q (p^{2} + q^{2})$

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Solution

Prove
$(p + q)^{4} - (p - q)^{4} = 8 p q (p^{2} + q^{2})$
LHS
$= ((p + q)^{2})^{2} - ((p - q)^{2})^{2}$
Apply $a^{2} - b^{2} = (a + b) (a - b)$
$= ((p + q)^{2} + (p - q)^{2}) ((p + q)^{2} - (p - q)^{2})$
$= (2 (p^{2} + q^{2})) (4 p q)$
$= 8 p q (p^{2} + q^{2})$
= RHS
LHS=RHS

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P \lor \sim Q

\sim (\sim P \land Q)

(P \land Q) \lor (P \land \sim Q) \lor (\sim P \land \sim Q)

(P \land Q) \lor (P \land \sim Q) \lor (\sim P \land Q)

p_{1} < p_{2} < p_{3} < p_{4}

q_{1} < q_{2} < q_{3} < q_{4}

p_{4} - p_{1} = 8

q_{4} - q_{1} = 8

p_{1} > 5

q_{1} > 5

30

p_{1} - q_{1}

p^{4} - (p - q)^{4}

{(\frac{p}{q})}^{m} = \frac{q^{4}}{p^{4}}

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