How do you factor #3x ^ { 4} + 12x ^ { 3} + 4x + 24#?

1 Answer
Jan 25, 2018

Answer:

A few thoughts...

Explanation:

The quartic as it stands has no rational factors.

If there is a typo in the question and one of the terms is in error, then it may be possible to factor by grouping.

For example:

#3x^4+12x^3+4x+color(red)(16) = 3x^3(x+4)+4(x+4) = (3x^3+4)(x+4)#

#3x^4+12x^3+color(red)(6)x+24 = 3x^3(x+4)+6(x+4) = 3(x^3+2)(x+4)#

#3x^4+color(red)(18)x^3+4x+24 = 3x^3(x+6)+4(x+6) = (3x^3+4)(x+6)#

#color(red)(2)x^4+12x^3+4x+24 = 2x^3(x+6)+4(x+6) = 2(x^3+2)(x+6)#

It is possible to factor each of the above possibilities further using irrational coefficients.

For example:

#a^3+b^3 = (a+b)(a^2-ab+b^2)#

So:

#x^3+2 = x^3+(root(3)(2))^3 = (x+root(3)(2))(x^2-root(3)(2)x+root(3)(4))#

Alternatively, if the question is correct as given, then it is possible to factorise, but the answer gets very messy and complicated. I may write up a separate answer to sketch the process.