Factoring by pulling out a common factor

Statement

Factor: (a) $x^5 - 9x$; (b) $25r^3 - 16r$; (c) $12u^4 - 27u^2$; (d) $30v^3 + 18v^2 - 48v$.

Solution

(a) Pull out $x$ and recognize a difference of squares: $x^5 - 9x = x(x^4 - 9) = x(x^2 - 3)(x^2 + 3).$

(b) Pull out $r$: $25r^3 - 16r = r(25r^2 - 16) = r(5r - 4)(5r + 4).$

(c) Pull out $3u^2$: $12u^4 - 27u^2 = 3u^2(4u^2 - 9) = 3u^2(2u - 3)(2u + 3).$

(d) Pull out $6v$, then factor the trinomial $5v^2 + 3v - 8$ (two numbers with sum $3$ and product $-40$: they are $8$ and $-5$): $30v^3 + 18v^2 - 48v = 6v(5v^2 + 3v - 8) = 6v(5v + 8)(v - 1).$

$\boxed{x(x^2-3)(x^2+3)\ ;\ r(5r-4)(5r+4)\ ;\ 3u^2(2u-3)(2u+3)\ ;\ 6v(5v+8)(v-1)}$