108t^2-60t factor the polynomial
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Since each vector is a member of 
, the vectors will span if they form a basis for , which requires that they be linearly independent of one another.
To show this, you have to establish that the only linear combination of the three vectors
that gives the zero vector 
occurs for scalars 
.
Solving this, you'll find that, so the vectors are indeed linearly independent, thus forming a basis for and therefore they must span .
To show this, you have to establish that the only linear combination of the three vectors
Solving this, you'll find that
well, once you take in all the Revenue from sales, and subtract all the costs that went in for the product, what's leftover is the Profit.
so then, profit P(x), is simply the revenue minus costs, namely P(x) = R(x) - C(x)