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: \(\displaystyle (3x\,-\,12)(x\,+\,2)\), . . . incorrect
\(\displaystyle \;\;\)but how do you get any other answers?
The instruction 'Factor" always means "Factor
completely."
First, take out any common factors: \(\displaystyle \,3x(x^2\,-\,2x\,-\,8)\)
Then factor the trinomials: \(\displaystyle \,3x(x\,+\,2)(x\,-\,4)\)
Now it's factored completely.
\(\displaystyle \;\;\)(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 ...
\(\displaystyle \;\;\)there are
eight "right answers":
\(\displaystyle 3x^3\,-\,6x^2\,-\,24x\;=\)
\(\displaystyle \;\;3x(x\,+\,2)(x\,-\,4)\)
\(\displaystyle \;\;3(x^2\,+\,2x)(x\,-\,4)\)
\(\displaystyle \;\;3(x\,+\,2)(x^2\,-\,4x)\)
\(\displaystyle \;\;x(3x\,+\,6)(x\,-\,4)\)
\(\displaystyle \;\;x(x\,+\,2)(3x\,-\,12)\)
\(\displaystyle \;\;(3x^2\,+6x)(x\,-\,4)\)
\(\displaystyle \;\;(x\,+\,2)(3x^2\,-\,12x)\)
\(\displaystyle \;\;\)and, of course: \(\displaystyle \,1\,\times\,(3x^3\,-\,6x^2\,-\,24x)\)
It should be obvious why we should factor
completely.
