Question

When factored completely, the expression p^4 - 81 is equivalent to
(1) (p^2 + 9)(p^2 - 9)
(2) (p^2 - 9)(p^2 - 9)
(3) (p^2 + 9)(p + 3)(p - 3)
(4) (p + 3)(p - 3)(p + 3)(p - 3)

Answer 1

p⁴ - 81
p⁴ + 9p² - 9p² - 81
p²(p²) + p²(9) - 9(p²) - 9(9)
p²(p² + 9) - 9(p² + 9)
(p² - 9)(p² + 9)
(p² + 3p - 3p - 9)(p² + 9)
(p(p) + p(3) - 3(p) - 3(3))(p² + 9)
(p(p + 3) - 3(p + 3))(p² + 9)
(p - 3)(p + 3)(p² + 9)

The answer is (3).

Answer 2

we have

$p^4 - 81$

we know that

$(a^{2} -b^{2} )=(a+b)(a-b)$

so

$(p^4 - 81)=(p^{2} +9)(p^{2} -9)$ ------> expression $1$

and

$(p^{2} -9)=(p +3)(p-3)$ ------> expression $2$

Substitute expression $2$ in expression $1$

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

therefore

the answer is the option

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