How do you factor # x^4 + 2x^3 - 8x -16#?

2 Answers
Jun 27, 2018

Answer:

#(x+2)(x-2)(x^2+2x+4)#

Explanation:

#x^4+2x^3-8x-16#

#:.=x^3(x+2)-8(x+2)#

#:.=(x+2)(x^3-8)#

#:.=(x+2)(x^3-2^3)#

#:.=(x+2)(x-2)(x^2+2x+2^2)#

#:.=(x+2)(x-2)(x^2+2x+4)#

Jun 27, 2018

Answer:

Some are more straight forward to spot than others. Some you just have to try things out.

#(x+2)(x-2)(x^2+2x+4) #

Explanation:

Given: #x^4+2x^3-8x-16#

First observation is that #2xx8=16# so we have a possible connection there.

Lets have a look at the direct grouping:

#[color(white)(2/2)x^4+2x^3color(white)(2/2)]+[color(white)(2/2)-8x-16color(white)(2/2)]#

Consider the first brackets. If we factor out #x^3# we end up with:

#x^3[color(white)(2/2)x+2color(white)(2/2)]+[color(white)(2/2)-8x-16color(white)(2/2)]#

We can make the second brackets the same as the first if we factor out #(-8)# giving:

#x^3[color(white)(2/2)x+2color(white)(2/2)]-8[color(white)(2/2)x+2color(white)(2/2)]#

We now factor out the #(x+2)# giving:

#(x+2)(x^3-8)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Trying to factor the part "(x^3-8)#

This is the same as #(x^3-2^3)#

Can we manipulate a quadratic out of this? Lets have a play with:

#(x-2)("something")=x^3-8#

#color(blue)("Dealing with the "x^3" term")#

To obtain #x^3# we try: #(x-2)(x^2+?)=x^3-2x^2+?#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Dealing with the "x^2" term")#
but there is no #-2x^2# term in #x^3-8# so we need to get rid of it.

Try: #(x-2)(x^2+x+?)=x^3+x^2-2x^2-2x+? color(red)(" Fail")#

Try: #(x-2)(x^2+2x+?)=x^3+2x^2-2x^2-4x+? color(green)(" Works!")#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Dealing with the "-4x" term")#

#(x-2)(x^2+2x+?)=x^3-4x+?#
but there is no #x# term so we need to get rid of it.

Try: #(x-2)(x^2+2x+2)=x^3-4x+4xcolor(red)(-4) larr color(red)(" Fail")#

We got rid of the #x^2# term but ended up with the wrong constant.

Try: #(x-2)(x^2+2x+4)=x^3-4x+4x+8 larrcolor(green)(" Works")#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Putting it all back together")#

#(x+2)(x^3-8) color(white)("d")->color(white)("d")(x+2)(x-2)(x^2+2x+4) #

/////////////////////////////////////////////////////////////////////////////////////////////////////
#color(blue)("Bottom line comment")#

If you end up with #(a^3-b^3)# this becomes:

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

Where #color(white)("d")-(-b)a=+ab#