I'm confused by this Truth or False statement below:

If $7/4$ is a zero of some polynomial function f, then (4x-7) is a factor of f

Apparently, it is true, but since $7/4$ is a zero, we know that $(x-7/4)$ is a factor for sure, but how can we be sure that there is also a $4$ that can be multiplied to the said factor to make it (4x-7)?

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    $\begingroup$ "Factor" in what ring? $\endgroup$ – dxiv Jul 31 '17 at 7:42
  • $\begingroup$ @dviv Good question, but note that it is clearly a situation in which division by $4$ is possible. $\endgroup$ – Mark Bennet Jul 31 '17 at 7:44

$$f(x) = (x - \frac{7}{4})g(x)$$ is the same as $$f(x) = \frac{1}{4}(4x - 7)g(x)$$

Thus $4x - 7$ is a factor of $f(x)$.


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