We are given `P(t)=t^4-5t^2+4 ` as a model for the profits earned, where t is the time in years since 2005. We are asked to find the years with zero profit.
This is equivalent to solving the equation `t^4-5t^2+4=0 ` .
Note that this is a quadratic form -- it is a quadratic in `t^2 ` .
`t^4-5t^2+4=(t^2-4)(t^2-1) `
Each of the binomials is a difference of two squares and can be factored as:
`=(t+2)(t-2)(t+1)(t-1)...
We are given `P(t)=t^4-5t^2+4 ` as a model for the profits earned, where t is the time in years since 2005. We are asked to find the years with zero profit.
This is equivalent to solving the equation `t^4-5t^2+4=0 ` .
Note that this is a quadratic form -- it is a quadratic in `t^2 ` .
`t^4-5t^2+4=(t^2-4)(t^2-1) `
Each of the binomials is a difference of two squares and can be factored as:
`=(t+2)(t-2)(t+1)(t-1) `
So (t+2)(t-2)(t+1)(t-1)=0
`t=pm1,pm2 `
Since the domain is positive (the model is true for years after 2005), t=1 is 2006 and t=2 is 2007.
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The profit was zero in 2006 and 2007
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There were 4 solutions, but 2 of them (-1 and -2) were not in the domain and so are disregarded. (If the company came into existence in 2005, it makes no sense to describe its profits before 2005.)
The graph for t>0:
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