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)
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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+?
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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!")
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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")
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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)
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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
