1.
Consider the polynomial \(p(x) = x^4 + 5x^3 + x^2 + 3x + a_0\) (for some constant \(a_0\)). If you know that \(p(7)\) is a multiple of \(7\text{,}\) what can you conclude about \(a_0\text{?}\)
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Worksheet Rational Roots
Let’s play with some polynomials with integer coefficients and see if we notice anything interesting.
2.
Let \(p(x) = x^4 + 5x^3 + x^2 + 3x + 10\text{.}\)
(a)
Can you say whether \(x-7\) will be a factor without doing long division?
(b)
For which \(\alpha\) is it reasonable to check whether \(x - \alpha\) is a factor of \(p(x)\text{?}\)
3.
I’m thinking of a degree \(n\) polynomial \(p(x)\) with coefficients in \(\Z\) and \(\frac{5}{7}\) is a root of \(p(x)\text{.}\) What can we say about (at least some of) the coefficients of \(p(x)\text{?}\)
(a)
Write down the general form of \(p(x)\) using coefficients \(a_0, a_1, \ldots, a_n\text{.}\) Then write down an equation we know is true because \(\frac{5}{7}\) is a root.
(b)
Nobody likes fractions! Rewrite your equation from the previous step so that there are no fractions.
(c)
One of the terms in the equation will have no \(7\) in it. “Solve” for this term. What does that tell you about the other side of the equation, specifically involving 7? Can you conclude anything about \(a_n\text{?}\)
(d)
Now repeat the previous step, this time solving for the term containing no \(5\text{.}\) What can you conclude about \(a_0\text{?}\)
4.
The previous facts don’t depend on \(5\) and \(7\text{;}\) any \(s\) and \(t\) that have no common factors will work just as well. State a theorem that we have basically proved with the work above.
