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# Write the conjunctive or disjunctive normal form of an expression $f$

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Problem Detail:

Write the conjunctive or disjunctive normal form of an expression $f$ that is true if the number $t=(xyz)_2$ is a zero of the following polynomial:

$$p(t)=(t-6)(t-5)(t-4)(t-2)t(t+1)(t+3)(t+7)$$

The zeros of this polynomial are $-7,-3,-1,0,2,4,5,6$. So $t$ has to be $(000)_2$ or $(010)_2$ or $(100)_2$ or $(101)_2$ or $(110)_2$ or... But now I don't know how to represent these negative numbers in binary? In other words how do I write $-7,-3,-1$ in the form $(xyz)_2$?

As @DavidRicherby said, it's best if you ask who-gave-you-the-question what he meant by $(xyz)_2$. But if you absolutely can not do that, then you look at the situation mathematically as...

$(xyz)_2$ here is a set $\{000,001,010,011,...,111\}$ of $8$ elements which forms a ring. In particular, it supports two binary operation $+$ and $\times$ where both operation has identity element, the addition has inverses...

If you can decide the addition table and multiplication table that makes sense in your context, then you're set. As long as it's a ring, then that polynomial equation makes sense.