Question

factorise the equation
p⁴-81​

Answer 1

Answer:

(p²+9)(p+3)(p-3)

Step-by-step explanation:

=>p⁴-81

=>(p²+9)(p²-9)

=>(p²+9)(p+3)(p-3)

Answer 2

$\mathbb{\bold{\underline{ANSWER}}}$

$\fbox{\textsf{(p ^{\textsf{2}} + 9)(p + 3)(p - 3)}}$

$\mathbb{\bold{\underline{EXPLANATION:}}}$

$\textsf{p ^{\textsf{4}} - 81} \to \textbf{(p ^{\textbf{2}}) ^{\textbf{2}} - 9 ^{\textbf{2}} }$

  $\bullet \textbf{ Using the identity : a ^{\textbf{2}} - b ^{\textbf{2}} = (a + b)(a - b)}$

$\star \texttt{ a = p ^{\texttt{2}} }$

$\star \texttt{ b = 9}$

$\to \textsf{(a + b)(a - b) = (p ^{\textsf{2}} + 9)(p ^{\textsf{2}} - 9)}$

$\to \to \textbf{p ^{\textbf{4}} - 81 = (p ^{\textbf{2}} + 9)(p ^{\textbf{2}} - 9)}$

$\textsf{p ^{\textsf{2}} - 9 \to } \textbf{(p) ^{\textbf{2}} - 3 ^{\textbf{2}} }$

  $\bullet \textbf{ Using the identity : a ^{\textbf{2}} - b ^{\textbf{2}} = (a + b)(a - b)}$

$\star \texttt{ a = p}$

$\star \texttt{ b = 3}$

$\to \textsf{(a + b)(a - b) = (p + 3)(p - 3)}$

$\star \textsf{ So now, we have 3 factors : (p ^{\textsf{2}} + 9) , (p + 3) , (p - 3) }$

$\therefore \textbf{p ^{\textbf{4}} - 81 = (p ^{\textbf{2}} + 9)(p + 3)(p - 3)}$