Hello, brendaursula!
Factor: \(\displaystyle \:3x^3\,-\,6x^2\,-\.24x\)
the teacher said that there were three different answers that could be true.
. . . no!
i got one of them:
(3x−12)(x+2),
. . . incorrect
but how do you get any other answers?
The instruction 'Factor" always means "Factor
completely."
First, take out any common factors:
3x(x2−2x−8)
Then factor the trinomials:
3x(x+2)(x−4)
Now it's factored completely.
(The order of the factors makes no difference.)
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I hope you teacher realizes that it was a silly thing to say to a math class.
By his/her mind-set, there are
four ways to factor 24:
\(\displaystyle \;\;\;24\;=\;1\,\times\,24,\;\;2\,\times\,12,\;\;3\,\time\,8,\;\;4\,\times\,6\)
But there is only
one way: \(\displaystyle \,24\;=\;2\,\times\,2\,\time\,2\times\,3 \;=\;2^3\cdot3\)
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Even if your teacher is correct ... and we
don't have to factor completely ...
there are
eight "right answers":
3x3−6x2−24x=
3x(x+2)(x−4)
3(x2+2x)(x−4)
3(x+2)(x2−4x)
x(3x+6)(x−4)
x(x+2)(3x−12)
(3x2+6x)(x−4)
(x+2)(3x2−12x)
and, of course:
1×(3x3−6x2−24x)
It should be obvious why we should factor
completely.