How do you factor #27x ^3 + 216#?

Question

Isaiah Anthony · Accepted Answer

#27x^3+216 = 27(x+2)(x^2-2x+4)# Both terms in the original expression have a factor of #27# which can be extracted as a factor of the expression. The remaining factor #(x^3+8)=(x^3+2^3)# is of the form #color(white)("XXX")(x^3+a^3)# which we know has a factors #(x+a)# and #(x^2-ax+a^2)# or in this case #color(white)("XXX")(x^3+8)=(x+2)(x^2-2x+4)# By checking the discriminant of #(x^2-2x+4)# #color(white)("XXX")(-2)^2+4(1)(4) < 0 rArr # no further Real roots

Ethan Adkins · Answer

undefined To factor the expression 27x^3 + 216, you first find the greatest common factor (GCF), which is 27. Then, you rewrite the expression as 27(x^3 + 8). Now, you can factor the expression x^3 + 8 using the sum of cubes formula, which is (a^3 + b^3) = (a + b)(a^2 - ab + b^2). Applying this formula, you get x^3 + 8 = (x + 2)(x^2 - 2x + 4). So, the factored form of 27x^3 + 216 is 27(x + 2)(x^2 - 2x + 4).